Method and apparatus for characterizing composite materials using an artificial neural network

ABSTRACT

This invention relates to a method and apparatus for characterizing composite materials, and in particular, to utilizing an artificial neural network for predicting an impact resistance of a composite material. A method for predicting an impact resistance of a composite material in accordance with the present invention includes the steps of designing an artificial neural network including a plurality of neurons, training the artificial neural network to predict the impact resistance by adjusting an output of the plurality of neurons according to sample data and known results of the sample data, inputting data of the composite material into the artificial neural network, and utilizing the artificial neural network to predict the impact resistance of the composite material.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to a method and apparatus for characterizingcomposite materials, and in particular, to utilizing an artificialneural network for predicting an impact resistance of a compositematerial.

2. Description of the Related Art

The “background” description provided herein is for the purpose ofgenerally presenting the context of the disclosure. Work of thepresently named inventor, to the extent it is described in thisbackground section, as well as aspects of the description which may nototherwise qualify as prior art at the time of filing, are neitherexpressly or impliedly admitted as prior art against the presentinvention.

Composite materials have been in human use in different forms forthousands of years, examples of earlier use of composite materials maybe seen in the mud and straw bricks.

Composite materials for construction, engineering and other similarapplications are formed by combination of two or more materials in orderto enjoy the benefits of the properties of the constituents. A propertyof composite materials is that the materials are still distinguishableand don't blend completely unlike alloys, hence, normally exhibit aninterface between one another. The constituent materials retain theirphysical and chemical properties, only to combine to give propertiesthat are not offered by the individual constituents.

The majority of composite materials use two constituents: a binder ormatrix and reinforcement. The reinforcement is stronger and stiffer,forming a sort of backbone, while the matrix keeps the reinforcement ina set place. The binder also protects the reinforcement, which may bebrittle or breakable.

As illustrated in FIG. 1, composites may be categorized in three maindivisions according to the geometry of the reinforcements: (1)Particle-reinforced, (2) Fiber-reinforced, and (3) StructuralComposites.

According to the type of the matrix, there are: (1) Polymer MatrixComposites, (2) Metal Matrix Composites, and (3) Ceramic MatrixComposites.

Technologically, important composites may be those in which thedispersed phase is in the form of a fiber. Design goals offiber-reinforced composites often include high strength and/or stiffnesson a weight basis. In fiber-reinforced composites, fibers are the phasethat provides the strength and the ability to carry load while thematrix increases the ductility and also acts as binding agent for thefibers and also acts as load transfer medium.

Common fiber reinforcing agents include, Aluminum, Aluminum oxide,Aluminum silica, Asbestos, Beryllium, Beryllium carbide, Berylliumoxide, Carbon (Graphite), Glass (E-glass, S-glass, D-glass), Molybdenum,Polyamide (Aromatic polyamide, Aramid), e.g., Kevlar 29 and Kevlar 49,Polyester, Quartz (Fused silica), Steel, Tantalum, Titanium, Tungsten,Tungsten monocarbide.

Common resin materials include Epoxy, Phenolic, Polyester, Polyurethene,and Vinyl Ester.

Composite pipes are gradually replacing the conventional pipes in theindustrial applications. Composite pipes show good resistance tocorrosion compared to metallic pipes in applications where pipes arecarrying fluids like water or highly corrosive sulphuric acid is presentin it. This property makes them ideal for usage in pipe industry [37].

Composite pipes may be described in two categories depending upon thetype of resin material: (1) Reinforced thermosetting resin pipes (RTRP),and (2) Reinforced thermoplastic pipes (RTP).

Due to their superior mechanical and thermal properties overconventional materials, fiber reinforced composite materials arepreferred in the petroleum industry. As an example of the advantagegained by replacing conventional material pipelines with compositematerials is that a 6-inch diameter pipe weighs 4 pound per foot,whereas copper nickel pipe with the same diameter weighs 24 pound perfoot [51].

Another major area of significant interest where composite pipes may beof use is the water related applications. Lack of fresh water reservoirsput forward the need of desalination applications. The desalinationapplication requires piping systems that are corrosion resistant [61].Water losses due to degradation of traditional pipe systems present asignificant financial and maintenance problem. Composite based pipingsystems provide good protection against the corrosion. Fiberglass pipesystems have become the material of choice in the desalination and waterdistribution industry.

There are several major advantages composite pipes offer overconventional material pipes, such as corrosion resistance. Fiberglasspipes are resistant to corrosion for a long period of time and resistscorrosion to a variety of media including seawater, hot brine, acids andother chemicals [61]. Also, the composite materials have a high strengthto weight ratio compared to metals and the transportation andinstallation of the composite materials is easier. Large lengths ofcomposite pipes may be easily manufactured and may be assembled withrelative ease on sites.

Since, composite materials are corrosion resistant; the cost ofmaintenance is considerably lower. Also, the fatigue resistantcapability of composite pipes is better than the metallic pipes. Also,low internal friction, fire resistance, torsional stiffness and goodimpact resistance combined with the flexibility in design as perstrength and other requirements make them ideal replacement for thecurrent conventional materials [61].

Mechanical damages to pipes occur frequently. These damages may causeleakage of oil and gas from pipes resulting from structural failure andmay lead to reduced operating pressure or stopped production, human andenvironmental hazards and the heavy economic losses [7].

There are, however, some issues related to the use of composite pipingsystems primarily the lack of test data to support the materials' longterm durability. The failure caused by the mechanical damages is one ofthe important aspects that need to be addressed. The structural failureof these pipelines may be due to a number of effects as burst, impact,puncture, overload, buckling, fatigue and fracture.

One of the major causes of damages in pipes are considered as “ExternalDamage” caused by foreign objects and third party damage such as causedby a farmer ploughing a drainage ditch, or a supply boat dragging itsanchor around an offshore platform [24]. These structural components areoften very susceptible to foreign object impact during service. Thesedamages may be vulnerable and may go unseen especially in case of lowvelocity impacts since these are not visually observable. A small dentcaused by such impacts may lead to significant underlying damages forexample, delamination, matrix cracking, fiber breakage and fiber/matrixinterfacial debonding induced within the laminate [27].

Outside forces are one of the major causes of pipeline failures.Historically, the pipelines used were made from steels. Steel is aductile material and the specifications used in the industry are alreadyset for its use. The ASME codes B3 1.4 for oil applications and B31.8for gas applications provide measures for the different kind of damagesand repairs [12]. These materials are tested for their ductile behavior.Impact tests are considered good method to measure toughness ofpipelines.

During the product lifecycle it is always expected that damages mayoccur due to impact by foreign objects. Mechanical damage may occurduring handling, installation and service to the composite pipes. Toensure the reliability, good impact properties against low andintermediate velocity impacts are needed. Due to the laminate structureof composite materials their behavior to impacts is different to themetallic structures. The modes of damage in composite structures due toimpact may be categorized as matrix cracking, fiber breakage and/ordelamination [14].

Impact generally causes low to medium energies which cause a globalstructural response, and often results in internal cracking anddelamination, while at higher energy levels may cause penetration andexcessive local shear damage [1].

The impact damage may be caused by a number of factors, some of whichare for example:

Dropped tool

Damage due to mishandling

In-service impacts

Hail and debris

The composite materials are prone to low energy impacts that may beobserved with the effect of delamination in the plies and may beindirectly responsible for the failure. Delamination result in loweringof the elastic moduli, strength, durability and damage tolerance [14].Low velocity impacts may also cause matrix cracking which sometimes maynot be on the surface of impact but on the internal or bottom surface,this is due to the fact that the laminate is flexible. Matrix crackingis in the perpendicular direction to the plane of the laminate and is atensile crack. In thicker laminates, matrix cracking is near the topsurface and characterized as the shear crack.

The damage in composite materials due to impact force is a complexmechanism and still there are no analytical methods that may begenerally accepted to define the phenomenon.

In addition to these, the micro failure modes commonly observed incomposite laminates are fiber breakage, fiber micro buckling and matrixcrushing, transverse matrix cracking, transverse matrix crushing,debonding at the fiber-matrix interface and delamination [14].

SUMMARY OF THE INVENTION

The foregoing paragraphs have been provided by way of generalintroduction, and are not intended to limit the scope of the followingclaims. The described implementations, together with further advantages,will be best understood by reference to the following detaileddescription taken in conjunction with the accompanying drawings.

Disclosure of the inventor, Muhammad Haris Malik, “Optimization ofImpact Resistance of composite Plates and Pipes,” Thesis, King FahdUniversity of Petroleum & Minerals, Dhahran, Saudi Arabia, December,2012, is hereby incorporated in its entirety. Additionally, allreferences addressed in this disclosure are hereby incorporated in theirentireties.

By studying in detail the available literature, a number of motivationshave been found to continue the work in the field of optimization of theimpact resistance of composite laminated plates and pipes. It isapparent that a lot of effort by various researchers around the globehas been put into the study of the behavior and dynamic response ofcomposite materials under low velocity impact loading. Most of the workhas been focused on the damage characterization and the initiation andpropagation of damage under certain conditions. These studies haveprovided a great insight into the behavior and response of compositelaminates plates and shells when impacted by foreign objects havinglow-velocity impacts. While there have been a lot of parametric studiesconsidering the effects of various factors involving both the compositestructure and the impactor, there is no logical conclusion to theeffects which enhances the impact resistance of such structures. It isknown from these studies that the impact response of composite platesdepend upon the size, shape, mass and velocity of the impactor, also theimpact response is the characteristic of the material and geometricproperties of the composite plate or shell itself. This is apparent thatthe properties and circumstances involving the impactor are not in thecontrol of the designers; rather the composite plates or shells may bemanipulated such that the impact performance of these structures may beenhanced.

The studies provide a general understanding of different effectsmaterial, geometric and boundary conditions of the composite structurehave on the impact resistance. This provides the opportunity to furthertake these studies and develop such characteristics of materials andother factors related directly to the composite structure so that theimpact performance may be increased.

An exemplary implementation of the present invention may include amethod for predicting an impact resistance of a composite material. Sucha method may comprise designing an artificial neural network including aplurality of neurons, training, performed by a processor, the artificialneural network to predict the impact resistance by adjusting an outputof the plurality of neurons according to sample data and known resultsof the sample data, inputting data of the composite material into theartificial neural network, and utilizing the artificial neural networkto predict the impact resistance of the composite material.

In such a method, the artificial neural network may include an inputlayer of neurons that receives data that is input into the artificialneural network, and an output layer of neurons that outputs theprediction of the impact resistance of the composite material. Theartificial neural network may further include a hidden layer comprisinga plurality of neurons, the hidden layer may receive data output fromthe input layer, and the hidden layer may output processed data to theoutput layer.

Training the artificial neural network may include inputting the sampledata to the input layer, measuring an error between the known results ofthe sample data and the prediction output from the output layer, andreducing the error by managing the hidden layer such that data outputfrom neurons in the input layer may be selected for input to individualneurons in the hidden layer, and applying a variable weighting factor toeach neuron of the plurality of neurons in the artificial network toadjust an output of each neuron.

In such a method, training the artificial neural network may includeinputting the sample data to the artificial neural network, measuring anerror between the known results of the sample data and the predictionoutput from the artificial neural network, and reducing the error byapplying a variable weighting factor to each neuron of the plurality ofneurons in the artificial neural network to adjust an output of eachneuron. The error may be a mean-squared error.

In such a method, the input data of the composite material may includeany of the following: a stacking sequence of layers in the compositematerial; a layer thickness; a number of layers in the compositematerial; an orientation angle of the layers in the composite material;and a material composition of the layers in the composite material.

In such a method, the artificial neural network may be a feed forwardnetwork.

In another exemplary implementation of the present invention, a devicemay be utilized to predict an impact resistance of a composite material.Such a device may comprise a processor configured to design anartificial neural network including a plurality of neurons, train theartificial neural network to predict the impact resistance by adjustingan output the plurality of neurons according to sample data and knownresults of the sample data, input data of the composite material intothe artificial neural network, and utilize the artificial neural networkto predict the impact resistance of the composite material.

In such a device, the artificial neural network may include an inputlayer of neurons that receives data that is input into the artificialneural network, and an output layer of neurons that outputs result datafrom the artificial neural network.

In such a device, training the artificial neural network may includeinputting the sample data to the artificial neural network, measuring anerror between the known results of the sample data and the predictionoutput from the artificial neural network, and reducing the error byapplying a variable weighting factor to each neuron of the plurality ofneurons in the artificial neural network to adjust an output of eachneuron.

Such a device may be a component in a system for predicting an impactresistance of a composite material that is in accordance with anexemplary implementation of the present invention.

In another exemplary implementation of the present invention, anon-transitory computer readable medium may store computer readableinstructions that, when executed by a computer, may cause the computerto perform a method that includes designing an artificial neural networkincluding a plurality of neurons, training the artificial neural networkto predict the impact resistance by adjusting an output of the pluralityof neurons according to sample data and known results of the sampledata, inputting data of the composite material into the artificialneural network, and utilizing the artificial neural network to predictthe impact resistance of the composite material.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 illustrates a chart that categorizes composites according togeometry of the reinforcements.

FIG. 2 illustrates a flowchart of a methodology for optimizing compositeplates and pipes.

FIG. 3A illustrates a schematic drawing showing the geometric dimensionsof the composite plate.

FIG. 3B illustrates a schematic drawing of a nose tip of an impactor.

FIG. 4 illustrates a layup plot and material orientation of thecomposite plate.

FIG. 5A illustrates a schematic drawing showing the geometric dimensionsof a composite pipe.

FIG. 5B illustrates a layup plot and material orientation of a compositepipe.

FIG. 6 illustrates a graph of linear damage evolution.

FIG. 7A illustrates boundary conditions on a full plate model.

FIG. 7B illustrates boundary conditions on the quarter plate model.

FIG. 7C illustrates boundary conditions on the composite pipe model.

FIG. 8A illustrates a mesh configuration for the full composite platemodel.

FIG. 8B illustrates a mesh configuration for the quarter plate model.

FIG. 8C illustrates a mesh configuration for the composite pipe model.

FIG. 9 illustrates mesh convergence for dissipated energy with respectto maximum displacement.

FIG. 10A illustrates maximum displacement for the composite plate withrespect to time.

FIG. 10B illustrates a displacement contour at the instant of 1.6 msecat a kinetic energy level of zero.

FIG. 11A illustrates mesh convergence with respect to maximum Von-Misesstress.

FIG. 11B illustrates mesh convergence with respect to maximumdisplacement.

FIG. 11C illustrates mesh convergence with respect to maximum peakforce.

FIG. 11D illustrates mesh convergence with respect to rebound velocityof the impactor.

FIG. 12 illustrates a comparison force vs. time plot.

FIG. 13A illustrates a block diagram of a nominal system.

FIG. 13B illustrates a block diagram of a perturbed system.

FIG. 14A illustrates a graph of normalized sensitivity coefficients forvariables demonstrating a relative effect of each absorbed impactenergy.

FIG. 14B illustrates a graph of normalized sensitivity coefficients forvariables having a greater influence on the amount of absorbed energyexcept thickness.

FIG. 15A illustrates a comparison of carbon and glass composite platesat varying thicknesses with stacking sequence 1.

FIG. 15B illustrates a comparison of carbon and glass composite platesat varying thicknesses with stacking sequence 2.

FIG. 15C illustrates a comparison of carbon and glass composite platesat varying thicknesses with stacking sequence 3.

FIG. 15D illustrates a comparison of carbon and glass composite platesat varying thicknesses with stacking sequence 4.

FIG. 16A illustrates scatter data for layer configuration 1 forcarbon/epoxy plates.

FIG. 16B illustrates scatter data for layer configuration 2 forcarbon/epoxy plates.

FIG. 16C illustrates scatter data for layer configuration 3 forcarbon/epoxy plates.

FIG. 16D illustrates scatter data for layer configuration 4 forcarbon/epoxy plates.

FIG. 17 illustrates a force vs. time plot of CFRP plates of twodifferent thicknesses using the [0/30/60/90] laminate configuration.

FIG. 18A illustrates scatter data for layer configuration 1 forglass/epoxy plates.

FIG. 18B illustrates scatter data for layer configuration 2 forglass/epoxy plates.

FIG. 18C illustrates scatter data for layer configuration 3 forglass/epoxy plates.

FIG. 18D illustrates scatter data for layer configuration 4 forglass/epoxy plates.

FIG. 19A illustrates a force vs. time plot of GFRP plates of twodifferent thicknesses using [45/−45/0/90] laminate configuration.

FIG. 19B illustrates a force vs. time plot of GFRP plates of twodifferent thicknesses using [45/−45/0/90] laminate configuration withfracture energy of 40 kj/m².

FIG. 20A illustrates absorbed energy vs. thickness for stackingsequences 1-4 for carbon/epoxy systems.

FIG. 20B illustrates a comparison of absorbed energy for stackingsequences for thin CFRP plates.

FIG. 20C illustrates absorbed energy vs. thickness for stackingsequences 1-4 for carbon/epoxy systems.

FIG. 20D illustrates a comparison of absorbed energy for stackingsequences for thick GFRP plates.

FIG. 21A illustrates a comparison of an amount of absorbed energy forCFRP plates with 16 layers and CFRP plates with 20 layers.

FIG. 21B illustrates a comparison of an amount of absorbed energy basedon a number of layers for GFRP plates with a fixed thickness.

FIG. 21C illustrates a comparison of GFRP plates with an increase inperformance and an increase in layers.

FIG. 21D illustrates a comparison of GFRP plates with a decrease inperformance and an increase in layers.

FIG. 22A illustrates absorbed energies for CFRP and GFRP pipes with an35° winding angle.

FIG. 22B illustrates absorbed energies for CFRP and GFRP pipes with an45° winding angle.

FIG. 22C illustrates absorbed energies for CFRP and GFRP pipes with an55° winding angle.

FIG. 22D illustrates absorbed energies for CFRP and GFRP pipes with an65° winding angle.

FIG. 22E illustrates absorbed energies for CFRP and GFRP pipes with an75° winding angle.

FIG. 23A illustrates absorbed energy vs. thickness of a plate for 35°winding angle GFRP pipes.

FIG. 23B illustrates absorbed energy vs. thickness of a plate for 45°winding angle GFRP pipes.

FIG. 23C illustrates absorbed energy vs. thickness of a plate for 55°winding angle GFRP pipes.

FIG. 23D illustrates absorbed energy vs. thickness of a plate for 65°winding angle GFRP pipes.

FIG. 23E illustrates absorbed energy vs. thickness of a plate for 75°winding angle GFRP pipes.

FIG. 23F illustrates absorbed energy vs. thickness of a plate for 35°winding angle CFRP pipes.

FIG. 23G illustrates absorbed energy vs. thickness of a plate for 45°winding angle CFRP pipes.

FIG. 23H illustrates absorbed energy vs. thickness of a plate for 55°winding angle CFRP pipes.

FIG. 23I illustrates absorbed energy vs. thickness of a plate for 65°winding angle CFRP pipes.

FIG. 23J illustrates absorbed energy vs. thickness of a plate for 75°winding angle CFRP pipes.

FIG. 24A illustrates absorbed energy for CFRP pipes with winding anglesof 35°-75°.

FIG. 24B illustrates absorbed energy for GFRP pipes with winding anglesof 35°-75°.

FIG. 25A illustrates variations in absorbed energy with respect towinding angle for CFRP pipes.

FIG. 25B illustrates variations in absorbed energy with respect towinding angle for GFRP pipes.

FIG. 26A illustrates absorbed energy in equal thickness plates with avarying number of layers for 35° winding angle GFRP pipes.

FIG. 26B illustrates absorbed energy in equal thickness plates with avarying number of layers for 45° winding angle GFRP pipes.

FIG. 26C illustrates absorbed energy in equal thickness plates with avarying number of layers for 55° winding angle GFRP pipes.

FIG. 26D illustrates absorbed energy in equal thickness plates with avarying number of layers for 65° winding angle GFRP pipes.

FIG. 26E illustrates absorbed energy in equal thickness plates with avarying number of layers for 75° winding angle GFRP pipes.

FIG. 26F illustrates absorbed energy in equal thickness plates with avarying number of layers for 35° winding angle CFRP pipes.

FIG. 26G illustrates absorbed energy in equal thickness plates with avarying number of layers for 45° winding angle CFRP pipes.

FIG. 26H illustrates absorbed energy in equal thickness plates with avarying number of layers for 55° winding angle CFRP pipes.

FIG. 26I illustrates absorbed energy in equal thickness plates with avarying number of layers for 65° winding angle CFRP pipes.

FIG. 26J illustrates absorbed energy in equal thickness plates with avarying number of layers for 75° winding angle CFRP pipes.

FIG. 27A illustrates absorbed energy vs. plate thickness forcombinations of varying composite layers and woven carbon lamina.

FIG. 27B illustrates absorbed energy vs. the position of the carbonlayer for a 5.8 mm plate.

FIG. 27C illustrates absorbed energy vs. the position of the carbonlayer for a 11 mm plate.

FIG. 28A illustrates absorbed energy vs. pipe wall thickness of GFRP andCFRP pipes with different combinations of woven carbon fabric layers.

FIG. 28B illustrates absorbed energy vs. pipe wall thickness of hybridCFRP and GFRP pipes with different layer orientations.

FIG. 29 illustrates a graphical representation of a single neuron in anartificial neural network.

FIG. 30 illustrates a multi-layered feed forward artificial neuralnetwork.

FIG. 31A illustrates a correlation between a predicted response and atarget response for CFRP plates.

FIG. 31B illustrates scatter data of an actual response and vs. apredicted response for CFRP plates.

FIG. 32A illustrates a correlation between a predicted response and atarget response for GFRP plates.

FIG. 32B illustrates scatter data of an actual response and vs. apredicted response for GFRP plates.

FIG. 33A illustrates a correlation between a predicted response and atarget response for CFRP pipes.

FIG. 33B illustrates scatter data of an actual response and vs. apredicted response for CFRP pipes.

FIG. 34A illustrates a correlation between a predicted response and atarget response for GFRP pipes.

FIG. 34B illustrates scatter data of an actual response and vs. apredicted response for GFRP pipes.

FIG. 35 illustrates a flowchart of a method in accordance with anexemplary implementation of present invention.

FIG. 36 illustrates an apparatus in accordance with an exemplaryimplementation of the present invention.

DETAILED DESCRIPTION OF THE DRAWINGS

Referring now to the drawings, wherein like reference numerals designateidentical or corresponding parts throughout the several views.

FIG. 2 illustrates a flowchart of a methodology for optimizing compositeplates and pipes. In particular, the optimization of the compositeplates and pipes is divided into two phases, in the initial phase amodel is developed for the composite laminated plates and the study isbased upon models and results from available literatures.

Experimental Studies on Plates and Laminates

There have been a number of studies on the effect of differentparameters on the impact characteristics of composite plates and pipes.These studies include experimental [2,5,13,25,58,63,66,69] numerical[6,40,41,66], and analytical [26] which discuss the impact behavior ofdifferent composite laminates and discuss the effects of variousparameter changes and a number of studies which studied numerically[32,34,35,56,72,73] and a few experimental studies [43,72] have alsobeen performed on laminated composite shells. There are a number ofstudies which have developed analytical or numerical techniques to studythe impact response of composite plates and shells under low energyimpact damage.

Yang & Cantwell [71] conducted a number of low velocity impact tests on(0°,90°) glass fiber reinforced epoxy resin to study the effects ofvarying key parameters on the damage initiation threshold. The resultsshow that the impact resistance is proportional to the thickness of thecomposite panel. Also, the tests show that the impact resistance was notaffected by the plate's geometry. A further study by Yang et al was doneto study the effect of impactor shape. The focus of their study was theeffect of key parameters, such as target size, projectile diameter andtest temperature on damage initiation. The tests were carried out onsamples of unidirectional E-glass fiber reinforced FM94 epoxy resin. Themajority of tests were conducted on laminates of 1.8 mm thickness whilefew tests were carried out on laminates of thickness ranging from 0.8 mmto 3.6 mm. Tests were also undertaken to study the effect of temperatureon the damage initiation. Tests were carried out at temperatures of 45,60, 75 and 90° C. In these tests, the damage initiation threshold wasestablished by increasing the impact energy until delamination justbecame apparent in the test samples. The samples were not subjected tomultiple impact tests considering that would result in fatigue and alower value of damage threshold. The tests conducted by Yang andCantwell, suggested that the damage initiation force is proportional tothe target thickness. The tests demonstrated dependency in the order oft3/2, where ‘t’ is the thickness of the composite plate. This result wasverified with the studies conducted earlier. They also carried outexperimental studies on whether the geometry of the test specimeneffects on the damage initiation threshold. This result was alsosupported by earlier studies that the damage initiation threshold doesnot depend upon the panel size. The final parameter studied was theeffect of temperature and it was expected that temperature will have aneffect on the matrix fracture. Tests were conducted at a number oftemperatures between 23° and 90° C. A linear relationship was observedbetween the thickness of the panel and the damage initiation force at aparticular temperature. It was observed that the damage thresholdincreased with temperature for thinner laminates.

Ker{hacek over (s)}ys, Ker{hacek over (s)}ienė, & {hacek over(Z)}iliukas [5] studied the impact response of woven carbon/epoxy andE-Glass/epoxy composite systems on vehicle body structures byconsidering energy profile diagrams and force-displacement curves. Forlow velocity impact tests, drop weight tests were performed. Todetermine the mechanism of impact damage the experiment was performedwhen laminated composite materials were deformed with low impact energy.The maximum energy used in the test was equal to 120 J by means of avertically falling impactor. The total amount of energy introduced to acomposite specimen and the energy absorbed by the composite specimenthrough the impact event are important parameters to assess impactresponse of the composite structures. The experiments demonstrated thefact that was also displayed by numerical studies was that the reductionin the stiffness of the composite plate. To estimate the energy absorbedduring the impact a contact force F(t) was measured during the impact.This force depends upon the impactor mass ‘m’ and the velocity ‘v’.Given an initial velocity ‘v0’, that is a function of acceleration dueto gravity and downfall height ‘H’.v ₀=√{square root over (2gH)}

Impactor speed and displacement ‘s’ as the function of time are given byintegrating the impact force:

${v(t)} = {v_{0} - {\left( \frac{1}{m} \right){\int_{0}^{t}{{F(t)}\ {\mathbb{d}t}}}}}$${s(t)} = {\int_{0}^{t}{\left( {v_{0} - {\left( \frac{1}{m} \right){\int_{0}^{t}{{F(t)}\;{\mathbb{d}t}}}}} \right).}}$

The kinetic energy of the impactor and the absorbed energy

$E_{imp} = {\frac{1}{2}{mv}^{2}}$${E_{ab}(t)} = {{\frac{1}{2}{mv}_{0}^{2}} - {\frac{1}{2}{{m\left( {v_{0} - {\left( \frac{1}{m} \right){\int_{0}^{t}{{F(t)}\ {\mathbb{d}t}}}}} \right)}^{2}.}}}$

It was observed that the stiffness of E-Glass/Epoxy composites duringimpact decreased with the increasing displacement due to great specimendeflection related with non-linear membrane effect. Force-timerelationships were almost symmetrical. But the area under theforce-displacement curve showed the great part of impact energy absorbedwith the laminar composite at low velocity impact energies. The resultsshow that at low impact energies of 6 J, the force value of 3.08 kN wasmaximum which gradually decreases to zero. But when the impact energy isgreater, the maximum force value is reached when the damage under theimpactor occurs after the greater total displacement.

Rilo & Ferreira [58] conducted their study on the experimentalinvestigation of low velocity impacts on glass-epoxy laminated compositeplates. The characterization of the damage was done in relation to thetype of test, stacking sequence, dimensions and the maximum force of theimpact.

Numerical Studies for Impact on Composite Plates

A number of studies were also carried out using the numerical approachto investigate the impact response of composite laminates and plates.

Setoodeh et al. [62] used a three dimensional elasticity based approachcoupled with the layer wise laminated plate theory by J. N. Reddy. Thestudy considers the effects of low velocity impact of general fiberreinforced laminated composite plates. A custom finite element code wasdeveloped for the impact response based on 3-D elasticity approach.Hertzian nonlinear contact law used to model the contact forces betweenthe impactor and the target surface. The effect of impact velocity, massof the impactor and the material properties were studied. The methodapplied by Setoodeh et al adopts a combined two- and one-dimensionalanalysis, which reduces the number of manipulations and the complexityin the formulation of the 3-D finite element method. The procedure isnot completely three-dimensional yet it is capable of describing theimpact behavior economically and accurately at the same time. In the FEmodeling of Setoodeh et al, 9 noded quadratic surface elements with 3noded quadratic elements in the thickness direction were used.

Farooq & Gregory [18] developed a finite element computational model tostudy the impact behavior and the failure of CFRP panels that areimpacted with low velocity drop-weight. The impactor used for the studyis a flat-nosed tip object. Farooq et al used the commercially availablesoftware ABAQUS to study the critical damage regions under and near theimpact zone. In-plane stresses were calculated from the model and thetransverse shear stress were calculated using Trapezium rule from thestandard equilibrium equations. The method used in this study isdifferent from the Setoodeh et al as it is a 2-D model to predict the3-D transverse shear stress. The calculated and the predicted stresseswere used with failure theories to predict possible failure modes.

Farooq & Gregory [17] in the paper titled “Finite Element Simulation ofLow Velocity Impact Damage Morphology in Quasi Isotropic CompositePanels Under Variable Shape Impactors” studied the barely visible impactdamage (BVID), its initiation, growth and tolerance in fiber basedcomposites under the low velocity impact. The impact damage reduces thestiffness of the composite panel and this concept was used in the model.Quasi isotropic specimens were selected to model the damage in the fiberdirections. Three different specimens and three different types ofimpactor nose shapes were used. It is predicted that under the sameloading conditions different nozzle tips produce different damages. Theenergy absorbed during the impact is dissipated in the form of matrixdamage, fiber fracture and delamination, this result in significantlyreduced stiffness. Low velocity impacts mean longer contact time betweenimpactor and target surface which causes global deformation which maycause internal damage that may be difficult to detect. Farooq et al havestudied the effect of such damages on the stiffness and the operationallife of composite panels after low velocity impacts.

Tiberkak et al. [65] has investigated the response of Fiber reinforcedcomposite under the low velocity impact loads. Mindlin's plate theory isimplemented in the FE model which uses a 9-noded Lagrangian element. Thestudy suggests that the increase in 90 degree plies increase the contactforce implying a reduction in the rigidity of the laminate. Initially,threshold velocities were evaluated for matrix crack initiation.Afterwards, using appropriate failure criteria will be used to predictmatrix cracking at higher velocities. The results in this study suggestthat the damage occurs in the upper 90 degree plies with the dominanceof transverse shear stress. The study is based upon the impact of aspherical object with low velocity upon a composite laminated platecontaining a number of transversely thin layers and the contact force isapplied at the center of the plate. The Mindlin plate theory takes intoaccount the effect of transverse shear deformation and is applied inthis study. The impact between the impactor and the composite plate isconsidered frictionless, the damping in the plate is neglected and theimpactor is considered as a rigid body with isotropic properties. Thisstudy also applies the Hertzian law to calculate the contact forcebetween the impactor and the composite plates. The study performs aparametric analysis by varying boundary conditions, stacking sequence,size of the composite plate and velocity of the impactor.

Tiberkak et al. observed no significant variations in the results withthe change in boundary conditions. The effect of change of the stackingsequence shows that the contact force increases with the increase in thethickness of the 90 degree plies that mean the rigidity of the laminatesis reduced. The contact forces increase with an increase in thepercentage of fibers in the 90 degrees direction.

Heimbs et al. [22] conducted their analysis of impact on a compositeplate with compressive preloads. Since, in real life systems thecomposite plates may be subjected to different stress states when it isbeing impacted and hence its behavior may be different from the unloadedor without stress behavior. The main issues covered by Heimbs et al arethe modeling of composite laminate, its delamination and theimplementation of preload. Impact loads are considered as a transientload and hence FE codes are based on explicit time integration, usingsmall time step intervals. But the preloading is a static load makingthe use of implicit calculations more appropriate. That's why Heimbs etal have used specific numerical techniques for the solution of acombination of preloading and impact loadings. The results of the studywere supported by a number of tests conducted on the drop weight testmethod. The tests were conducted for both preloaded and unloadedcomposite laminates. The tests conducted on compressive preloadedspecimens indicated that preloading results in increased deflection ofthe CFRP plates and hence more material damage. This is due to the factthat more energy is absorbed and less is rebounded as elastic springback effect which is the case with unloaded CFRP plates. The FE modelwas developed in LS-DYNA and it was developed with the modeling of thecomposite material including the intra laminar failure and delaminationfailure, the modeling of the preload and the impactor. The compositelaminate was modeled as 24 plies of unidirectional laminas as 2-D shellelements. A number of failure criteria were defined based on the loadingand the material damage such as tensile failure in matrix direction,tensile failure in fiber direction, compressive failure in matrixdirection and compressive failure in the fiber direction. Failure isconsidered as soon as one of these criteria was met. In addition tothese, strain based failure was also defined.

Interlaminar failure is another major phenomenon in the low velocityimpacts of composite laminates, delamination absorbs energy upon impactand as a result the stiffness of the laminate is reduced. In LS-DYNA,there are two methods to include delamination as described by [22]. Oneof the methods is to use the cohesive brick elements between separatelayers of shell elements with material law that may describe the damageprocess of the laminate connection.

The literature survey showed that so far the majority of the work in theimpact analysis of composite materials has been focused on the study ofcomposite laminates and very few studies have considered compositeshells such as pipes. There is a lot of potential in the researchrelated to the impact response of composite pipes and need to developsolutions for the improvement of impact characteristics of compositepipes subjected to low velocity impacts.

Naik and Meduri [53] studied the effect of laminate configuration on theimpact behavior of composite laminates. Studies were carried out ondifferent mixed composites, cross-ply laminates, woven-fabric compositesand 3-D composites. The studies concentrated the effect of differentlaminate configurations on the impact response. The impactor mass,velocity and the incident impact energy were kept constant keeping inview of the typical tool drop scenario. From the study it is observedthat the mixture of Unidirectional and woven fabrics demonstrates moreresistance to impact damage.

Studies on Composite Shells

A limited number of studies have also been done on the impact behaviorof composite shells.

Ibekwe et al. [27] discussed the effect of a thin metallic shell bondedto the outer surface of a laminated composite shell as a bumper layer.The experimental study revealed that the inclusion of a thin aluminumsheet increased the initiation energy that is the metallic sheet wasable to absorb some of the impact energy. The maximum impact load andthe deflection at maximum load were increased and the impact durationreduced. The higher impact loads did not cause considerable damage inthe specimens with bonded aluminum sheet and only a slight reduction inthe bending strength of the specimen was observed compared to thespecimen without the aluminum sheet. The study by Ibekwe et al. showedthat the damage was primarily in the bumper layer i.e. the aluminumsheet and it has served its purpose of absorbing the impact energy.

Yokoyama, Donadon, & de Almeida [72] presented an energy based failuremodel to study the impact resistance of the composite shell laminates.The damage model is formulated using a combination of stress based,continuum damage mechanics and fracture mechanics approaches within aunified procedure by using a smeared cracking formulation. The damagemodel was implemented in ABAQUS as a user defined material for shellelements and the damage model was validated with experimental resultsfrom previously available studies. In total five failure criterions wereused in the study namely, tensile and compression fiber failure, tensileand compression matrix cracking and in-plane shear failure modes definedas:

$\begin{matrix}{{Tensile}\mspace{14mu}{fiber}\mspace{14mu}{failure}} & {\frac{\sigma_{11}}{X_{t}} \geq 1} \\{{Compression}\mspace{14mu}{fiber}\mspace{14mu}{failure}} & {\frac{\sigma_{11}}{X_{c}} \geq 1} \\{{Tensile}\mspace{14mu}{matrix}\mspace{14mu}{cracking}} & {\frac{\sigma_{22}}{Y_{t}} \geq 1} \\{{Compression}\mspace{14mu}{matrix}\mspace{14mu}{cracking}} & {\frac{\sigma_{22}}{Y_{c}} \geq 1} \\{{In}\text{-}{plane}\mspace{14mu}{shear}\mspace{14mu}{failure}} & {\frac{\tau_{12}}{S_{12}} \geq 1}\end{matrix}$

Based on these failure criterions, damage evolution laws were developedfor fiber breakage and matrix cracking. They studied the effects ofthree parameters namely the presence of pressure loading, the laminatethickness and curvature. The main contribution of the paper is thedevelopment of damage models and the verification. The numerical resultsindicated that thickness, curvature and pressure significantly affectthe damage extent on pressurized composite laminates under impactloading. This becomes more visible for plates, which shows a greatersusceptibility to the pressure effects. The damage extent under impactloading decreases when combined with internal pressure effects. Theresults indicated that larger the plate curvature higher is the amountof dissipated energy during the impact loading. Moreover, the amount ofdissipated energy decreases as the plate thickness increases.

Her et al. [23] studied the effects of low velocity impacts on shellstructures using ANSYS/LS-DYNA as well as the effect on the compositelaminates. The effects of parameters like shell curvature, type ofsupport boundary conditions and impactor velocity were analyzed. Theresults show that the structures which have smaller curvature andclamped boundary condition result in a larger contact force and lessdeflection. In the study by Her et al., the focus was on the evaluationof transient response of the impact on composite laminates, cylindricaland spherical shells.

Krishnamurthy et al. [36] discussed the impact response and the damageof laminated composite shells by a metallic impactor using FiniteElement Method. The important parameters that formed the basis of studywere impactor mass and velocity, shell curvature and stacking sequence.Also, studied was the effect of presence of initial stress.

The paper by Pinnoji and Mahajan [56] presents a numerical study on theimpact resistance of composite shells laminates using energy basedfailure model. The damage model formulation is based on a methodologythat combines stress based, continuum damage mechanics (CDM) andfracture mechanics approaches. The damage model has been implemented asa user defined material model in ABAQUS FE code within shell elements.[56]

Krishnamurthy et al. [34] studied the impact response using theclassical Fourier series and the FEM. Impact response determined by thefinite element method also includes a prediction of the impact-induceddamage deploying the semi-empirical damage prediction model ofChoi-Chang. A parametric study was carried out by the finite elementmethod to determine the effect of varying the controlling parameterssuch as impactor mass, its approach velocity, curvature of the shell, onboth the impact response and on the impact-induced damage. A reductionof the stiffnesses of the failed laminas on the impact responseconcurrently as the solution proceeded has also been incorporated.

The study by Zhao et al. focuses on the impact-induced damage initiationand propagation for laminated composite shells under low velocityimpacts. The damage analysis is performed by using Tsai-Wu quadraticfailure criterion, Tsai's damage modes and additional delaminationformula at all Gaussian points. The damage modes considered are matrixcracking, fiber breakage and delamination. The progressive failure isexpressed by reducing stiffness of the material at all failed Gaussianpoints. The analyses of the flat and curved laminates are compared fordiscussing their different damage mechanism. In addition, the influenceof the stacking sequence, the thickness and the radius of curvature ondamage behavior of composite shells is studied [73].

Sensitivity Analysis

Sensitivity analysis is a tool employed in engineering problems toidentify the influence of input parameters on the state variables suchas displacements, stresses, strains and temperature etc. The result ofsensitivity analysis is the identification of a limited set of state orinput variables that have greater influence on the output of the system.The main aim of the sensitivity analysis is the calculation of thesensitivity coefficients [54] which is obtained by the variation ofinput variables one at a time or in groups and study the variation inthe output variable [57].

The sensitivity coefficient is computed by partially differentiating thestate function; defining the output; with respect to the inputparameters. These derivatives may be computed numerically using thebasic equations defining the system output or may be calculatedanalytically if a closed form solution exists. This sensitivitycoefficient may be calculated using analytical functions, also somecombined numerical and analytical methods for calculation are availablein the literature [19]. The computation of sensitivity coefficients issuggested to be normalized so that a direct comparison of all the inputvariables may be deduced. The actual benefit of normalized sensitivitycoefficient (NSC) is that it provides an information about the order ofmagnitude of variation in the output variable with the change of oneorder of magnitude in the input variables [47].

The methodology of using sensitivity analysis is a common practice infor almost all types of numerical techniques [31]; Boundary ElementMethod (BEM) [33], Finite Difference Method (FDM) [33], Finite ElementMethod (FEM) [9] as well as hybrid and meshless strategies [15,41]. Thistechnique provides a very helpful tool in narrowing down the complexvariables involved in the design of composite structures.

Finite Element Methods are one of the best developed numerical tools forthe structural analysis and the use of sensitivity analysis along withFEM has been quite common. In a study from 1993, Noor and Shah [54] usedthe technique to estimate the sensitivity coefficients of unidirectionalfiber-reinforced composites for the effective thermal and thermoelasticproperties.

The sensitivity analysis approach is successfully used in a wide rangeof applications. Bilal et al. [57] used the approach to identifyimportant model parameters in their study of evaporative coolers andcondensers. They use the normalized sensitivity coefficients to studythe effects of input variables that have the most influence on theresponse variables of the condensers and cooler systems. The method usedto calculate the normalized sensitivity coefficient in this study isbased upon the formulation presented by Bilal et al. in their paper,which will be discussed in detail later on.

Artificial Neural Networks

Artificial Neural Networks (ANN) models are a very powerful method sincethey may be applied to any generic problem with few inputs and may betrained to learn from them with the expected outputs. These networksmimic the behavior of the neurons inside a human brain and it is arguedthat even at 0.1% of its performance, it is still an extraordinaryprocessing system [29]. ANN models proved to be excellent tool in theapproximation and interpolation in a variety of applications[10,11,21,28,39,42,44-46,55,67,70]. ANN has been used in functionfitting and prediction of various mechanical properties and damagemechanisms in composite materials. ANN models are very efficient formodeling and predicting the non-linear behavior of different systems.

El Kadi [29] has presented a comprehensive review of the neural networksand the different approaches within them. ANNs are generally composed ofa number of neurons spread over few layers that are interconnected.These models are trained against some target data and response set andthe model are trained such that it is able to predict the output to acertain range of the training set. The progress is measured againsteither the mean-square error (MSE), root-mean-square error (RMSE), ornormal-mean-square error (NMSE) between the observed output and thetarget output. The applications of ANN are in the manufacturing processoptimization as well as in the monitoring and modeling the manufacturingand the mechanical behavior of fiber-reinforced composites. El Kadi haspresented a brief review of all the applications of ANN in the field offiber reinforced polymeric composites.

Bezerra et al. [11] used ANN to predict the shear stress-strain behaviorof carbon/epoxy and glass/epoxy fabric composites. The authors used themulti-layered neural network model and demonstrated that about 80% ofstandard error of prediction was ≧0.9. In their study, they consideredthe stress as a function of the orientation angle by layers, specimen offiber and the shear strain, while certain other factors like porosity,number of layers, matrix type and volumetric fraction of fibers were notstudied.

Vassilopoulos et al. [67] used ANN to model the fatigue life ofmultidirectional GFRP composite laminates. The benefit that ANN providedthe authors was the approach saved around 50% experimental effort forthe whole analysis as compared to conventional methods and that toowithout the loss of considerable accuracy. It is mentioned that theartificial neural networks are effective tools to model fatigue life ofcomposite materials and also to build the constant life diagrams. Theauthors have used the error back propagation (EBP) algorithm for thetraining of the neural network. The neural network used was a multilayerfeed forward network having four inputs namely θ (off axis angle), R(stress ratio), σ_(max) (maximum stress), and σ_(a) (stress amplitude).

Jiang et al [28] applied the ANN model to predict the mechanical andwear properties of the short fiber reinforced polyamide composites. Thepolyamide composites were reinforced by short carbon and glass fibersand then optimization of the neural networks was performed. The neuralnetwork was used to predict the mechanical and wear properties as afunction of the content of fibers and testing conditions. In this study,the authors have also used the back propagation neural networkalgorithm.

Design of Experiments

Design of experiments, or experimental design, is the design of allinformation-gathering exercises where variation is present, whetherunder the full control of the experimenter or not. The purpose of it isto study the effect of some processes or intervention on some objects.Design of experiment is a discipline which has broad applications acrossall the natural and social sciences. A methodology for designingexperiments was proposed by Ronald A. Fisher, in his innovative book TheDesign of Experiments (1935).

Design of experiments is a very efficient statistical technique whichmay be employed in various experimental investigations [3]. The designof experiments provides the capability to understand the design effectsof various factors and their statistical significance as well [50]. Thedesign of experiments is useful at the stage of data collection as itprovides a systematic and rigorous approach which generates valid,defensible and supportable data sets.

Design Optimization and Algorithms

Optimization is an integral part of design and is very beneficial forthe commercial production of structures. The ability design engineerspossess using composite materials is the custom made properties tailoredexactly according to the needs of the structures. But, the compositematerials involve more design variables compared to conventionalmaterials which make it difficult to optimize the design and achievemaximum performance. This difficulty induces the need to useoptimization techniques in the design process of composite materials.

Almeida et al. [4] used genetic algorithms for the design optimizationof the composite laminated structures. The authors have discussed theadaptation of the terminologies and developing codes to use them withGA. The technique is used to study multi-objective optimization ofplates under transverse or in-plane loads. The objectives of the studywere the weight and the cost or the deflection and weight.

Lee et al [38] have used evolutionary algorithms for the multilayeredcomposite structure design optimization. The objective of their studywas the optimization of the stacking sequence of the composite plates.The authors have shown that the optimal solutions have lower weight,higher stiffness and affordable costs compared to other cases. They alsodiscussed the benefits of parallel optimization systems.

Swaroop et al [68] used the optimization techniques to optimize the plyangles and the internal geometry of the helicopter rotor blades madeusing composite materials. The authors studied the multi objectiveoptimization of several conflicting objectives which included thestiffness parameters, blade mass and the distance between mass centerand the aerodynamic center of the blades. They discussed thetransformation of multi-objective optimization to a single optimizationproblem and then applying a Particle Swarm Optimization technique tofind the optimal solution.

Suresh et al. [64] also used the Particle Swarm Optimization for multiobjective optimization of the design of box beam made of compositematerials. The optimal solution was used to design a helicopter rotorblade. The ply angles and the cross-sectional area are considered thedesign parameters needed to optimize.

Numerical Model

Initially, a numerical model of a flat plate was developed in ABAQUSExplicit environment and used to verify against the available resultsfrom the literature. The inventor chose the model from the study ofYokoyama et al [72], the study by Yokoyama et al. was based uponexperimental and numerical results. The experimental results were basedupon the thesis of Biase EHC., and the same model was developed in theABAQUS to verify the model.

The numerical model was based on the same assumptions and materialmodels as the one for the composite flat plates. The results werevalidated for the filament wound composite pipes against theexperimental results available in the thesis by Mohammed Khaliq Naik[52]. The study by Naik was experimental and performed in the AdvancedMaterial Science Lab at King Fahd University of Petroleum and Minerals,and hence will be better correlated.

In the following sections, the basic parameters and characteristics ofthe numerical model for both the flat plates and pipes will be discussedsimultaneously.

Idealizations and Assumptions

The plate and the pipe are assumed to be a 2-D shell with the layersdefined in the composite section, while the impactor was considered as a3-D rigid element with a reference point (pilot node) defined at the tipof the impactor. The initial velocity was given to the reference pointof the impactor just before the impact as it is assumed to be under afree fall motion from a certain height achieving the velocity due togravitational acceleration.

The contact is assumed to be frictionless without loss of much accuracy.It is assumed that the kinetic energy of the impactor just before theevent of impact begins will be transferred to the specimen as the impactenergy and this will be transferred to the subject in the form ofinternal energy, the amount of increase in the internal energy should beequal to the amount of decrease in the kinetic energy of the impactor asit bounces back. The amount of damage caused to the specimen as a resultof impact will be evident from the amount of energy absorbed by theplate or the pipe. This energy absorbed will describe the damage to thecomposite specimen.

Geometric Model

In this research work, the impact performances of both composite platesand pipes have been studied. The composite plate model is modeled as thestudy of Yokoyama et al. [72], while the composite pipes were modeled asthe experimental setup of Naik [52].

Geometric Model for Composite Flat Plate

The geometric dimensions of the composite plate and the impactor andalso the stacking sequence of the plate are defined as:

TABLE 1 Geometric Dimensions of the composite plate and impactor formodel validation. Composite Plate Impactor Length 102 mm Diameter 12.7mm Width 152 mm Mass 1.5 kg Thickness  4.2 mm Velocity 6.0608 m/s

For the case of model validation, the laminate is consisted of 20 layersof equal thickness of 0.21 mm having the stacking sequence of[(±45)/(0,90)/(±45)/(0,90)/(±45)]_(2s). Initially, a full model wasdeveloped for the model validation purposes, which was then reduced toquarter symmetry to save the computational efforts. The results were notmuch affected with the quarter symmetry.

FIG. 3A illustrates a schematic drawing showing the geometric dimensionsof the composite plate. FIG. 3B illustrates a schematic drawing of anose tip of an impactor. As illustrated in FIG. 3B, only the nose tip ofthe impactor is modeled due to the reason that the impactor is assumedto be a rigid material and the study was not interested in the stressdistribution in the impactor. Therefore, it is appropriate to model onlythe nose tip of the impactor which comes into contact with the specimenand avoid the added complexity of the whole impactor geometry. The noseof the impactor has the dimensions as prescribed in the ASTM D2444standards.

FIG. 4 illustrates a layup plot and material orientation of thecomposite plate. The layers are defined as symmetric about the middleplane, and hence only the half number of layers are defined and usingthe option in ABAQUS of symmetric plies. The layers are defined suchthat the primary direction of fibers is coincident with the globalx-axis, these layers and the orientation may be visualized asrepresented in FIG. 4.

Geometric Model for Composite Pipes

The dimensions of the composite pipes were selected so that it may bevalidated with the experimental results from the thesis of MohammedKhaliq Naik [52]. These experiments and the thesis study were carriedout in the King Fahd University of Petroleum and Minerals and hence havea better correlation with the future experimental works if performed.Also, the dimensions are dictated by the ASTM Standards ASTM D2444.

According to the ASTM D2444 standards, the pipe length should be atleast equal to the nominal outside diameter but not less than 6 in. (152mm) [48]. Since, the diameter of the pipe in this case is 150 mm; thelength of the pipe is taken as twice the diameter as suggested.

FIG. 5A illustrates a schematic drawing showing the geometric dimensionsof a composite pipe. FIG. 5B illustrates a layup plot and materialorientation of a composite pipe.

TABLE 2 Geometric Dimensions of the Composite Pipe and the Impactor forthe Model Validation. Composite Pipe Impactor Length 300 mm Diameter12.7 mm Internal Diameter 150 mm Mass 10 kg Thickness  6 mm Velocity2.8284 m/s

The specimen is considered to be manufactured using filament windingtechnology, which generally winds fiber around a sand mandrel at aspecific angle. Since, the process of winding goes from end to end onthe mandrel, the winding angle varies from +θ to −θ. This kind of layersare defined in ABAQUS using the composite section without the usage ofsymmetric layers option as there is no mid-plane about which the layersare symmetric.

For the case of model validation, the winding angle is kept at 55° asreported in the work of Naik. The winding angle of 55° is a preferredchoice of winding angle among the industry as it is known to have goodperformance against both the axial loading and internal pressure [8].The number of layers is assumed to be 24 with each layer havingthickness of 0.25 mm, as this is the popular layer thickness fromavailable literature and the supplier's information in the market.

Material Modeling

Composite materials as explained in the introduction are anisotropicmaterial having different material properties in different directions.For a layered composite, it is considered to be orthotropic withmaterial properties in the fiber direction higher than the materialproperties in the two transverse directions. Most commonly, the materialproperties in the two transverse directions are considered to be equal,this kind of material is considered to be transversely isotropicmaterial. The material used in the study is either carbon fiberimpregnated with epoxy resin or glass fiber. Generally, flat plates areconstructed using woven fabrics and the pipes are manufactured using thefilament winding technology. The material properties and behavior istherefore, different as the woven fabric is usually available in theform of cross-ply woven form which makes it different from the layersfrom filament winding which is essentially a unidirectionalconstruction.

Hooke's law for transversely isotropic materials defines fiveindependent elastic constants, which are the Young's modulus andPoisson's ratio in the y-z symmetry plane, Young's modulus and Poisson'sratio in the perpendicular direction and the shear modulus in theperpendicular direction. The compliance matrix is given as the Eq.(3.1):

$\begin{matrix}{\begin{bmatrix}ɛ_{xx} \\ɛ_{yy} \\ɛ_{zz} \\ɛ_{yz} \\ɛ_{zx} \\ɛ_{xy}\end{bmatrix} = {\begin{bmatrix}\frac{1}{E_{x}} & {- \frac{v_{yx}}{E_{y}}} & {- \frac{v_{yx}}{E_{y}}} & 0 & 0 & 0 \\{- \frac{v_{yx}}{E_{x}}} & \frac{1}{E_{y}} & {- \frac{v_{zy}}{E_{y}}} & 0 & 0 & 0 \\{- \frac{v_{xy}}{E_{x}}} & {- \frac{v_{yz}}{E_{y}}} & \frac{1}{E_{y}} & 0 & 0 & 0 \\0 & 0 & 0 & \frac{1}{2\; G_{yz}} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1}{2\; G_{xy}} & 0 \\0 & 0 & 0 & 0 & 0 & \frac{1}{2\; G_{xy}}\end{bmatrix}\begin{bmatrix}\sigma_{xx} \\\sigma_{yy} \\\sigma_{zz} \\\sigma_{yz} \\\sigma_{zx} \\\sigma_{xy}\end{bmatrix}}} & (3.1)\end{matrix}$

An y-z plane was considered the plane of symmetry, Ey=Ez, vxy=vxz, andvyx=vzx. The symmetry of the stress and strain tensors dictates that:

$\begin{matrix}{{\frac{v_{xy}}{E_{x}} = \frac{v_{yx}}{E_{y}}},{v_{yz} = v_{zy}}} & (3.2)\end{matrix}$

However, both woven fabric and uni-directional laminates are consideredtransversely isotropic, a special subcategory of orthotropic materialsand following are the damage initiation models and the damage evolutionmodels for these materials.

Damage Initiation Modeling

Since the impact of the striker will cause damage, a damage model isneeded in order to describe when this damage begins and also once thedamage initiates how it will progress. In this study, the damageinitiation model as proposed by Hashin (1980) was used. The model asproposed by Hashin considers damage initiation in four different modes,namely,

Tensile Matrix Mode:

$\begin{matrix}{{{\frac{1}{Y_{t}^{2}}\left( {\sigma_{22} + \sigma_{33}} \right)^{2}} + {\frac{1}{S_{23}^{2}}\left( {\sigma_{23}^{2} - {\sigma_{22}\sigma_{33}}} \right)} + {\frac{1}{S_{12}^{2}}\left( {\sigma_{12}^{2} + \sigma_{31}^{2}} \right)}} \leq 1} & (3.3)\end{matrix}$

Compressive Matrix Mode:

$\begin{matrix}{{{{\frac{1}{Y_{c}}\left\lbrack {\left( \frac{Y_{c}}{2\; S_{23}} \right)^{2} - 1} \right\rbrack}\left( {\sigma_{22} + \sigma_{33}} \right)} + {\frac{1}{4\; S_{23}^{2}}\left( {\sigma_{22} + \sigma_{33}} \right)^{2}} + {\frac{1}{S_{23}^{2}}\left( {\sigma_{23}^{2} - {\sigma_{22}\sigma_{33}}} \right)} + {\frac{1}{S_{12}^{2}}\left( {\sigma_{12}^{2} + \sigma_{31}^{2}} \right)}} \leq 1} & (3.4)\end{matrix}$

Tensile Fiber Mode:

$\begin{matrix}{{\left( \frac{\sigma_{11}}{X_{t}} \right)^{2} + {\frac{1}{S_{12}^{2}}\left( {\sigma_{12}^{2} + \sigma_{31}^{2}} \right)}} \leq 1} & (3.5)\end{matrix}$

Compressive Fiber Mode:

$\begin{matrix}{\left( \frac{\sigma_{11}}{X_{t}} \right)^{2} \leq 1} & (3.6)\end{matrix}$

Where Yt, Yc, Xt, Xc represents the longitudinal tensile and compressiveand transverse tensile and compressive strengths respectively while S₁₂and S₂₃ represents the longitudinal and transverse shear strength.

Damage Evolution Model

Damage initiation is the event at which the initial damage is caused inthe laminate but once it is initiated this damage will progressivelyspread with further impact force. This is known as the damage evolutionand this will cause the strength of the composite laminate todeteriorate and hence the resulting product will be weaker compared toearlier before impact.

A simple energy based linear softening model is used as the damageevolution model. Energy damage evolution defines damage in terms of theenergy required for failure (fracture energy) after the initiation ofdamage. Linear softening specifies a linear softening stress-strainresponse for linear elastic materials or a linear evolution of thedamage variable with deformation for elastic-plastic materials.

For the damage initiation in plane stress fiber reinforced composites,the damage evolution law is available in ABAQUS; it assumes that beforedamage initiation the material was linearly elastic, with the stiffnessmatrix of a plane stress orthotropic material. After, the response ofthe material is computed from:σ=C _(d)ε  (3.7)

Where ε is the strain and Cd is the damaged elasticity matrix, given as:

$\begin{matrix}{C_{d} = {\frac{1}{D}\begin{bmatrix}{\left( {1 - d_{f}} \right)E_{1}} & {\left( {1 - d_{f}} \right)\left( {1 - d_{m}} \right)v_{21}E_{1}} & 0 \\{\left( {1 - d_{f}} \right)\left( {1 - d_{m}} \right)v_{21}E_{1}} & {\left( {1 - d_{m}} \right)E_{2}} & 0 \\0 & 0 & {\left( {1 - d_{s}} \right){GD}}\end{bmatrix}}} & (3.8)\end{matrix}$

Where D=1−(1−d_(ƒ))(1−d_(m))v₁₂v₂₁,d_(ƒ) gives the current state offiber damage, gives the current state of matrix damage and d_(s) givesthe current state of shear damage. The damage variables d_(ƒ), d_(m) andd_(s), are derived from damage variables d^(t) _(ƒ), d^(c) _(ƒ), d^(t)_(m), and d^(c) _(m) corresponding to the four failure modes describedfor Hashin model.

$\begin{matrix}{d_{f} = \left\{ {{\begin{matrix}d_{f}^{t} & {{{{if}\mspace{14mu}\sigma_{11}} \geq 0},} \\d_{f}^{c} & {{{{if}\mspace{14mu}\sigma_{11}} < 0},}\end{matrix}d_{m}} = \left\{ {{\begin{matrix}d_{m}^{t} & {{{{if}\mspace{14mu}\sigma_{22}} \geq 0},} \\d_{m}^{c} & {{{{if}\mspace{14mu}\sigma_{22}} < 0},}\end{matrix}d_{s}} = {1 - {\left( {1 - d_{f}^{t}} \right)\left( {1 - d_{f}^{c}} \right)\left( {1 - d_{m}^{t}} \right)\left( {1 - d_{m}^{c}} \right)}}} \right.} \right.} & (3.9)\end{matrix}$

FIG. 6 illustrates a graph that charts linear damage evolution.

Where G_(f) ^(t), G_(f) ^(c), G_(m) ^(t), G_(m) ^(c) and G_(s) are theenergies dissipated during damage for fiber tension, fiber compression,matrix tension, matrix compression and in-plane shear damage modesrespectively. The built-in damage evolution model in ABAQUS doesn'tsupport the in-plane shear damage.

Material Model for Composite Plates

The composite plates are manufactured using the woven fabric of carbonfiber or glass fiber impregnated with epoxy resin. The elastic materialproperties for the plates are listed in Table 3 for Carbon/Epoxy systemand in Table 4 for the Glass/Epoxy system.

The material properties used for the Carbon/Epoxy composite system istaken from the study of Yokoyama et al. [72] and is also used for thevalidation purposes. These values are quite close to the values cited inother literatures e.g. in the study by Pinnoji et al. [56], but asstated in the study by Yokoyama et al. the values are calculatedexperimentally. One of the points to note here is that the elasticmodulus in the z-direction demonstrated by subscript 3 is missing, butthis value has no consequence as the laminate material properties thataffect the overall solution are the in-plane properties and the materialproperties that ABAQUS requires are the laminate properties which doesnot include E₃. Also, generally this modulus is considerably lower thanthe moduli in the other two directions for the case of woven fabriccomposites.

TABLE 3 Mechanical elastic properties for orthotropic layer ofCarbon/Epoxy woven fabric used in composite plate modeling [72]. E₁ E₂E₃ G₁₂ G₁₃ G₂₃ (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) ν₁₂ ν₁₃ ν₂₃ 60.858.25 — 4.55 4.55 5 0.07 0.07 0.4

The material properties for the Glass/Epoxy system are selected from thestudy of Menna et al. [49].

TABLE 4 Mechanical elastic properties for orthotropic layer ofGlass/Epoxy woven fabric used in composite plate modeling [49]. E₁ E₂ E₃G₁₂ G₁₃ G₂₃ (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) ν₁₂ ν₁₃ ν₂₃ 26 26 8 3.82.8 2.8 0.1 0.25 0.25

The damage initiation as described here is defined in terms of thestress values compared to the strength of the lamina in a particulardirection under a particular loading condition. The strength values forthe Carbon/Epoxy are defined a Table 5 and for the Glass/Epoxy as Table6.

TABLE 5 Strength of composite layer in various directions forCarbon/Epoxy. X_(t) X_(c) Y_(t) Y_(c) S₁₂ S₂₃ (MPa) (MPa) (MPa) (MPa)(MPa) (MPa) Ply Strengths 621 760 594 707 125 125

TABLE 6 Strength of composite layer in various directions forGlass/Epoxy. X_(t) X_(c) Y_(t) Y_(c) S₁₂ S₂₃ (MPa) (MPa) (MPa) (MPa)(MPa) (MPa) Ply Strengths 414 458 414 458 105 65

The amount of damage due to the impact loads depend upon how the damagepropagates through the sample. The damage is said to initiate when thecritical strength limits were crossed and as more energy was applied bythe impactor the damage progressed through the sample. The amount ofenergy released or the amount of energy required to propagate the damagein the composite plate depends upon the intralaminar fracture energiesgiven in Table 7 for the Carbon/Epoxy system.

TABLE 7 Energy value for the damage evolution for Carbon/Epoxy. G_(f)^(t) G_(f) ^(c) G_(m) ^(t) G_(m) ^(c) G_(s) (KJ/m²) (KJ/m²) (KJ/m²)(KJ/m²) (KJ/m²) Intralaminar 160 25 10 2.25 2.25 Fracture Toughness

The fracture toughness is not available widely and if found most of theliterature studies only the critical value of the fracture toughnessthat is the value at which the damage or the crack initiates. For thisstudy, stress limit as the damage initiation and the use of energyrelease rates for modeling the propagation of damage was selected. Thevalue for the energy release rate in the fiber direction during tensionwas selected from the study of [16]. The rest of the values though haveless impact on the overall performance as will be shown in the latersections. Therefore, a simple ratio was adopted for the fracture energyin the fiber direction during compression and the matrix materials andis listed in Table 8.

TABLE 8 Energy value for the damage evolution for Glass/Epoxy. G_(f)^(t) G_(f) ^(c) G_(m) ^(t) G_(m) ^(c) G_(s) (KJ/m²) (KJ/m²) (KJ/m²)(KJ/m²) (KJ/m²) Intralaminar 10 1.562 0.625 0.14 0.14 Fracture Toughness

Material Model for Composite Pipes

The composite pipes are manufactured using the filament windingtechnology. This process of manufacturing pipes means that the layersare considered unidirectional lamina and hence the material propertiesand the plane of symmetry are different than the woven fabric. Theelastic material properties for the Carbon/Epoxy composite pipes areselected from the study of Yokoyama et al. [72] are listed in Table 9.

TABLE 9 Mechanical elastic properties for orthotropic layer ofCarbon/Epoxy unidirectional lamina used in composite plate modeling[72]. E₁ E₂ E₃ G₁₂ G₁₃ G₂₃ (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) ν₁₂ ν₁₃ν₂₃ 100 8.11 8.11 4.65 4.65 5 0.3 0.3 0.4

The elastic material properties for the Glass/Epoxy composite pipes areused from the study of Li et al. [41]. The model validation of the GFRP(Glass Fiber Reinforced Polymers) pipes was carried out with theexperimental study of Naik [52], but the thesis was mainly experimentaland all the material properties were not provided. Therefore, thematerial properties were calibrated and validated and it was found thatthe material properties given in the study of Li et al. [41] closelymatched the results. These material properties are tabulated in Table10.

TABLE 10 Mechanical elastic properties for orthotropic layer ofGlass/Epoxy unidirectional lamina used in composite plate modeling [41].E₁ E₂ E₃ G₁₂ G₁₃ G₂₃ (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) ν₁₂ ν₁₃ ν₂₃30.5 6.9 6.9 4.65 4.65 1.6 0.344 0.344 0.4

The strength properties of the Carbon/Epoxy lamina are given in Table 11and the strength properties of the Glass/Epoxy lamina are given in Table12.

TABLE 11 Strength of composite layer in various directions forCarbon/Epoxy. X_(t) X_(c) Y_(t) Y_(c) S₁₂ S₂₃ (MPa) (MPa) (MPa) (MPa)(MPa) (MPa) Ply Strengths 2000 1000 100 160 140 140

TABLE 12 Strength of composite layer in various directions forGlass/Epoxy. X_(t) X_(c) Y_(t) Y_(c) S₁₂ S₂₃ (MPa) (MPa) (MPa) (MPa)(MPa) (MPa) Ply Strengths 700 300 100 237 64 64

As it is described earlier, the damage propagation is modeled using theenergy release rates. These values for the CFRP are listed in the studyof Yokoyama et al. [72] and are listed in the Table 13. The intralaminarfracture toughness for the GFRP pipes are used from the study of Gershomand Marom [20].

TABLE 13 Energy value for the damage evolution for Carbon/Epoxy. G_(f)^(t) G_(f) ^(c) G_(m) ^(t) G_(m) ^(c) G_(s) (KJ/m²) (KJ/m²) (KJ/m²)(KJ/m²) (KJ/m²) Intralaminar 100 25 2 2 2 Fracture Toughness

TABLE 14 Energy value for the damage evolution for Glass/Epoxy. G_(f)^(t) G_(f) ^(c) G_(m) ^(t) G_(m) ^(c) G_(s) (KJ/m²) (KJ/m²) (KJ/m²)(KJ/m²) (KJ/m²) Intralaminar 52.5 20 2 2 2 Fracture Toughness

Loads and Boundary Conditions

This study is based on the damage caused due to the low-velocity impactloads. These loads are applied to the striker in the form of initialvelocity, which has kinetic energy equivalent to the amount of impactenergy intended to hit the specimen with. During experimentation, theimpact energy is controlled by the height from which the striker isdropped. The striker achieves the desired impact energy by virtue of thepotential energy transferred to the kinetic energy in the free fall.

Where, ‘m’ is the mass of the impactor, ‘h’ the drop height of theimpactor and ‘v’ is the velocity of the impactor just before it hits thetest specimen.

In ABAQUS, a reference point on the striker geometry is modeled and isgiven the mass and the velocity with which to impact the test specimen.

$\begin{matrix}{{{P.E.} = {mgh}}{{K.E.} = {\frac{1}{2}{mv}^{2}}}} & (3.10)\end{matrix}$

Case for Flat Plates

The impact load of 27.55 J was applied in the initial step of theexplicit dynamic analysis. This energy is provided to the striker ofmass 1.5 kg with an initial velocity of 6.0608 m/s.

FIG. 7A illustrates boundary conditions on a full plate model. Theboundary conditions are such that the shorter edges of the plate werefully constrained while the longer edges were set to be free. The impactenergy and the mass of the impactor and the boundary conditions are setaccording to the model from the study of Yokoyama et al. [72].

The quarter plate symmetry model was developed to reduce the size of theproblem, the loads were also reduced by ¼ which is achieved by dividingthe mass of the impactor by 4 such that the mass will be 0.375 kg. Toapply the symmetric boundary conditions, the two edges were constrainedto move in the direction of the axis of symmetry, as illustrated in theFIG. 7B.

Case for Composite Pipes

The boundary conditions for the impact analysis of composite pipes aredictated by the standards provided in the ASTM D2444. It is mentioned inthe standards that the pipe is supported with the help of a V-block. Thedesign of V-block should be such that it should be equal to the lengthof the pipe and has a 90° included angle. The support in the numericalmodel is provided at approximately the patches of the pipe where theV-block is supposed to be in contact with the pipe. The results in themodel validation proved that this simplification in the model wasaccurate.

FIG. 7C illustrates boundary conditions on the composite pipe model. Theimpact loads for the composite pipe are applied in the same way as forthe plates' impact analysis. The initial velocity is provided to thestriker which equates to 40 J of impact energy. The mass of the strikeris selected as 10 kg and the velocity to achieve the impact energy of 40J is 2.82843 m/s.

Element Type and Mesh

The composite plate and the pipe were modeled as the shell element,while the impactor was modeled as a rigid element. The element type S4Rwas used to mesh the composite plate and the pipes. The area near theimpact point was more refinely meshed rather than the whole model. It isobvious that the areas away from the impact point had less influence onthe numerical result. Hence, it was necessary to keep the mesh as coarseas possible in those regions so as to keep the number of nodes andelements to be solved to a minimum. This approach results in a highquality result with a much lesser amount of computational time spent.

FIG. 8A illustrates a mesh configuration for the full composite platemodel. FIG. 8B illustrates a mesh configuration for the quarter platemodel. FIG. 8C illustrates a mesh configuration for the composite pipemodel.

The mesh for the striker is not required as it is a rigid element andthe study was not interested in the deformation and stress in thestriker. A brief introduction about the element type used for thecomposite plate is discussed further.

Element Type—S4R

The thickness of the plate in this study is comparatively small than thelength and the width of the plate. For such structures, shell elementsare used. ABAQUS offers two types of shell elements, namely,conventional shell elements and the continuum shell elements.

S4R is a 4-node, quadrilateral, stress/displacement shell element withreduced integration and a large-strain formulation. This element is fromthe family of conventional shell elements and allows transverse sheardeformation and uses thick shell theory as the shell thickness increasesand become discrete Kirchhoff thin shell elements as the thicknessdecreases; the transverse shear deformation becomes very small as theshell thickness decreases.

This element type accounts for finite membrane strains and arbitrarilylarge rotations; therefore, they are suitable for large-strain analysisas in the case of impact analysis. Therefore, because finite strains andtransverse shear deformation are expected, S4R element has been chosenfor the simulation.

Model Validation and Sensitivity Analysis

Model validation is an important step of every numerical analysis. Ifthe numerical model is able to predict the results from the similarmodel from other studies either numerical or experimental, it gives theconfidence to use the model for the further analysis with the surety ofresults.

Model Validation of Composite Flat Plate

The model validation for the composite plates is carried out with thestudy by Yokoyama et al. [72]. The model geometry is described in Table1, which is the same as the model used in the study of Yokoyama. In thatstudy, Yokoyama et al. proposed a new damage initiation and evolutionmodel to better predict the impact damage and the energy absorbed duringthe impact event. For this study, the study started with the built-inmodel for the damage initiation and the damage evolution as describedearlier. It was found out that the results in this study are moreclosely matched from the results of the proposed model by Yokoyama etal. and also with the experimental results presented in their study.

Mesh Convergence

Mesh convergence is required to eliminate the numerical errors induceddue to finite element method which approximates the whole domain in afinite number of smaller elements. The results for the mesh convergenceare presented in the Table 15. The mesh was generated at two refinementlevels, with a refined central region where the impactor strikes thecomposite plate. The composite plate is meshed using the mapped meshingtechnique. Initially a constant element edge length of 3 mm was usedthroughout the plate which resulted in the generation of 2400 elementswith 2501 nodes, referred to as the refinement level 1 in the Table 15.The element edge length or edge seeds as better known in the ABAQUSenvironment were reduced to 2.5 mm for the refinement level 2.

At this point, the further reduction of element sizes would haveresulted in a large number of elements costing computational time, acentral region near the impact point was then refined further withoutreducing the edge lengths of the outer edges. In the first run, thecentral region edge length of elements was 1.25 mm and 2.5 mm for outeredges. The mesh at level 3 gave almost the double number of elements asprevious level with only about 3% improvement in the dissipated energyand less than 1% of the maximum displacement. However, a furtherrefinement was tried to make sure the convergence. Here to keep themapped meshing, the outer element edges were reduced to seed size of 2mm and central region to 1 mm.

TABLE 15 Mesh Convergence based on Maximum Displacement and DissipatedEnergy. Maxi- Re- Dissi- mum Dis- % age Dif- % age Dif- fine- patedplace- ference in ference in ment Ele- Energy ment Dissipated Displace-Level ments Nodes (J) (mm) Energy ment 1 1700 1785 6.167 6.2060 — — 22400 2501 5.0089 6.104 18.7 1.64 3 4704 4845 4.8674 6.062 2.82 0.688 47420 7597 4.7867 6.082 1.66 0.33

From the mesh convergence Table 15 and the related graph in FIG. 9, itis evident that the further refinement of mesh is not required and themesh at the refinement level 3 is sufficient. However, the refinementlevel 4 was preferred once the model was reduced to ¼ of the originalsize using the quarter symmetry model.

Validated Results

The results reported in the study by Yokoyama et al. are used tovalidate the model. This study reveals a much closer result to theexperimental values than the result from the proposed model. [72]

In Table 16 below, the results are shown for the experimental andnumerical results from the previous studies for both the Hashin modeland the model proposed by Yokoyama et al. and compared with the resultsfrom ABAQUS using the built-in Hashin model.

TABLE 16 Results from Yokoyama et al. and the comparison with theresults. Experi- Numerical mental Numerical (Hashin Our Error (Biase)(Yokoyama) Model) Result (% age) Maximum 0.006018 0.00611 0.005920.006062  0.7% Dis- placement (m) Time of 0.0036 0.00354 0.00328 0.003383.05% Impact Event (sec)

FIG. 10A illustrates maximum displacement for the composite plate withrespect to time. FIG. 10B illustrates a displacement contour at theinstant of 1.6 msec at a kinetic energy level of zero.

The results show that the maximum displacement occurs around 1.6 msec,after that the impactor bounced back with reduced velocity. This reducedvelocity resulted in the loss of kinetic energy which was absorbed asinternal energy in the composite plate. The kinetic energy of theimpactor as it bounces back reduces to just around 22.7 J which is equalto the amount of energy absorbed in damaging the plate.

There was very little difference in the values of the full plate and thequarter plate model as it is listed in Table 17.

TABLE 17 Comparison of results between full and quarter model. FullQuarter % age Model Model Difference Number of Elements 7420 2166   71%Number of Nodes 7597 2262   70% Dissipated Energy (J) 4.7867 4.743 30.91% Maximum Displacement (mm) 6.082 6.068 0.23% Time of Impact Event(msec) 3.38 3.32 1.78%

Model Validation of Composite Pipe

The model validation of the composite pipe case was done using theexperimental study conducted by Naik for his thesis work at MechanicalEngineering Department, King Fahd University of Petroleum and Minerals[52]. The geometry of the pipe and the impactor are already defined inChapter 3 Table 2. The material elastic properties, the strength valuesand the fracture energies are listed in Tables 10, 12 and 14respectively for the glass/epoxy system used in the study by Naik. Theboundary conditions are considered as defined by the ASTM standardsD2444.

In the study by Naik, they conducted the experiments at different energylevels for different pipe materials. This study selected the glass/epoxycomposite pipes under the impact load of 20 J for the case of modelvalidation. The geometric conditions and the loads are selected to besimilar to the experimental setup. The experimental study by Naikpresented the results in terms of peak force and therefore, this studybased its validation parameter to be the peak force rather than themaximum displacement as was the case with flat plate's model validation.

Mesh Convergence

Mesh convergence is an important aspect of finite element analysis. Itis necessary to refine the mesh to such a size that generates minimumamount of elements with a reasonable level of accuracy of results. Forthe composite pipes, the mesh convergence was carried out in two steps;initially a uniform mapped meshing was used throughout the pipe. Thiskind of meshing results in a very large number of elements costing a lotof computing time. To save the computing effort, a similar kind ofapproach was adopted as with the composite plates, that is, a finer meshin the central region where the impactor strikes the pipe and a morecoarse mesh outside. Initially, mesh convergence was carried out for areduced number of layers in order to save computational time required tosolve large number of integration points due to more layers. During themesh convergence, the total numbers of layers were considered to be 8with the winding angle of ±55°, each layer of 0.75 mm thickness.

For the uniform meshing, meshing was started with an element edge lengthof 10 mm and reducing it at each level where an element edge length wasreduced to just 2.2 mm. At this mesh refinement, as may be observed fromthe graphs illustrated in FIGS. 11A-11D most of the values have reachedthe constant value and further mesh refinement was not necessary. Fromthe Table 18, it may be noticed that at the seed level of 1.8 mm, thereis a sudden jump in the maximum displacement but further mesh refinementresulted in the displacement value to go the earlier level of around4.27 mm. Similarly, peak force also had one or two mesh levels where itincreased suddenly but overall it is constant around 7000 N.

TABLE 18 Mesh Convergence with Uniform Mesh Technique. Max Von Max PeakRebound Ele- Mises Disp Force Velocity Seed ments Nodes (MPa) (mm) (N)(m/s) 0.01 1536 1584 548 4.06 8163 1.70581 0.005 5700 5795 687 4.10 71571.67173 0.004 8816 8932 645 4.14 7229 1.65949 0.0025 23040 23232 7104.28 6950 1.6705 0.0022 28832 29040 690 4.26 7100 1.68511 0.002 3563635872 727 4.27 7090 1.70144 0.0018 43680 43940 692 5.92 7292 1.709840.0016 54896 55188 694 4.24 7523 1.73724 0.0015 63200 63516 693 5.977269 1.69964 0.0012 99396 99792 700 4.26 7106 1.67946 0.001 140400140868 695 4.18 7631 1.73527

As it is evident, from the Table 18 and the graphs showing meshconvergence that the refinement level with seed size of 2.2 mm havingmesh of about 30000 elements is appropriate for further studies. But, asshown in the graphs and Table 19, using the two level mesh refinementsis beneficial as it reduces the element numbers by almost half andwithout losing major accuracy. As a result, the further study wascarried out with the outer seed size of 4.4 mm and a 2.2 mm element edgelength for the central near the impact zone.

TABLE 19 Mesh Convergence with Non-Uniform Mesh Technique. Max Von MaxPeak Rebound Ele- Mises Disp Force Velocity Seed ments Nodes (MPa) (mm)(N) (m/s)  0.005-0.0025 12520 12640 696 4.34 6916 1.66958 0.0044-0.002213872 14008 698 4.26 7019 1.6841 0.004-0.002 16272 16416 709 4.27 69941.68893

Validated Results

The composite pipe model was validated with the experimental resultsfrom the thesis of Naik [52]. The validation was carried out against theimpact load of 20 J with a striker of 10 kg weight. The numerical modelrequires the material properties for the validation which were notprovided in the thesis. These values were then obtained from theavailable literature for similar kind of composite materials provided inthe study of Li et al. [40]. In the model validation phase, thegeometric dimensions were kept the same as reported in the work of Naik,but due to the fact that it doesn't provide the material properties aswell as the exact number of layers and the thickness of each layer. Anassumption was made considering the number of layers and thickness basedon the literature available on the subject. It was assumed that thelayers were 0.25 mm thick and the total numbers of layer were 24. Theresults are reported in Table 20.

TABLE 20 Model Validation results for the Glass/Epoxy Composite Pipe at20 J. Peak Force Deformation at Peak Time of Impact (N) Force (mm) Event(msec) Naik Thesis 6640 3.59 (12 J Impact) 7.4 Current Work 6970 3.95(20 J Impact) 7.76 % age Difference 4.9% — 4.86%

The results in Table 20 show that there is only about 5% differencebetween the results from Naik and the current numerical work, which is asufficient level considering the above justified assumptions. In thetable, the deformation at the peak force is mentioned for 12 J as therewas no value for deformation at the 20 J impact tests in the thesis byNaik.

In the graph illustrated in FIG. 12, the comparison between the forcevs. time plot is presented. One aspect that may be noticed that in thegraph displaying the Naik's result, it shows that after the first peakforce the force value remains less than the force values from thecurrent work. This may be explained on the basis that the sudden dropsoccurs when the force is such that it initiates the damage and hence inthe further contact the specimen is unable to offer more resistance. Onthe contrary, the peak force in the current research work reaches at thesame point of time but it didn't dip down below enough to meet theexperimental results. This may be due to the fact that the material usedfor experimental study has a lower tensile strength in the fiberdirection compared to the material properties used in the numericalresearch. Nevertheless, the results as shown are close enough and themodel may be safely validated.

Sensitivity Analysis

This study employs a sensitivity analysis approach to identify theparameters and quantitatively describe their degree of influence on theimpact resistance of the fiber reinforced polymer composite plates. Theresults were then used to optimize the factors in order to achieve thebest impact resistance for a certain case of composite laminate undercertain conditions of impact load and boundary conditions. The studiesprior to the current work have studied almost all the parameters indetail such as the thickness of the ply, stacking sequence and effect ofmaterials etc. as discussed in the literature review, but the currentfocus is to know how big the effect of one parameter is with respect toothers. This is needed in order to use the results in optimizationstudies where keeping the costs minimum is one criterion. It is wellestablished that increasing thickness and using stronger fiber materialincreases the impact performance but to optimize with cost in mind, itis important to know how to maximize the performance without increasingthe material costs. Hence, the needs to understand which parameter inaddition to thickness have greater effects. Also, once known whichmaterial properties have greater influence, it would be beneficial tosearch from the available materials with the least cost and bestproperties.

Sensitivity Analysis Formulation

In general, the sensitivity analysis is performed by varying one inputvariable at a time and observing its effect on the overall output. Let'sdenote the independent variables or the input variables with Xi and thevector X denotes the set of these variables.X= X±U _(x)  (4.1)

Where X denotes the nominal value of the independent variable and theU_(x) is the small

change about the nominal value. The range of ±U_(x) is defined such thatthe value of X may occur within this range with a certainty of about95%. Since the output parameter Y depends upon the input variables X, anuncertainty in X may be related to the output variable as:

$\begin{matrix}{U_{Y} = {\frac{\mathbb{d}Y}{\mathbb{d}X}U_{X}}} & (4.2)\end{matrix}$

Since, the input variable X is a vector of many different variables, theoutput variable Y must be a function of all the input variables suchthat;Y=Y(X ₁ ,X ₂ , . . . ,X _(N))  (4.3)

The uncertainty in Y may be expressed in terms of the root sum square ofall the individual uncertainties due to the input variables, that is;

$\begin{matrix}{U_{Y} = \left\lbrack {\sum\limits_{i = 1}^{N}\;\left( {\frac{\partial Y}{\partial X_{i}}U_{X_{i}}} \right)^{2}} \right\rbrack^{\frac{1}{2}}} & (4.4)\end{matrix}$

To normalize the sensitivity coefficients, one divides with the nominalvalue of the output

$\begin{matrix}{\left( \frac{U_{Y}}{\overset{\_}{Y}} \right) = \left\{ {\sum\limits_{i = 1}^{N}\;\left\lbrack {\left( {\frac{\partial Y}{\overset{\_}{Y}}\frac{{\overset{\_}{X}}_{i}}{\partial X_{i}}} \right)\left( \frac{U_{X_{i}}}{{\overset{\_}{X}}_{i}} \right)} \right\rbrack^{2}} \right\}^{\frac{1}{2}}} & (4.5)\end{matrix}$

The normalized sensitivity coefficient NSC is the term in the firstbracket on the right side of the equation, which is

$\begin{matrix}{{NSC}_{X_{i}} = \left( {\frac{\partial Y}{\overset{\_}{Y}}\frac{{\overset{\_}{X}}_{i}}{\partial X_{i}}} \right)^{2}} & (4.6)\end{matrix}$

This normalized sensitivity coefficient gives an opportunity to compareall the input variables and their effects with respect to one normalizedvalue of the nominal output variable. FIG. 13A represents the nominalsystem whose values and results are selected as reference and FIG. 13Bgraphically represents the variation in one parameter and its effect onthe output parameter.

For the sensitivity analysis of the structural problem, the generaloutput responses are the displacements, stresses, strains or velocities[30]. The output variable in the case of impact problem is taken asamount of energy absorbed during the impact event. This is considered asduring the impact event, the incident kinetic energy of the impactor istransferred to the specimen. This energy is absorbed in the form ofinternal energy of the specimen, which results in some of it used in theelastic deformation, some proportion of it used for plastic deformationof the fiber and epoxy while some of energy is dissipated in the damagemechanics as described by the Hashin Model. The energy used for theplastic deformation and damage is termed as the absorbed impact energyas it cannot be recovered while the energy stored in the elasticdeformation is returned to the impactor.

The sensitivity coefficients are computed numerically using the finiteelement method, the sensitivity analysis takes the amount of absorbedenergy as the output variable;{F}[K]{d}+[M]{{umlaut over (d)}}  (4.7)

The stiffness matrix is defined as:[K]=∫∫∫[B] ^(T) [D][B]dV  (4.8)

Where [D] is the material properties matrix and the matrix [B] is thegeometric properties matrix of the sample.

For a case of composite plate, the stiffness matrix may be given as;[K]=tA[B] ^(T) [D][B]  (4.9)

And the Mass matrix [M] is given by:

$\begin{matrix}{\lbrack M\rbrack = {\int_{V}{{{\lbrack N\rbrack^{T}\lbrack\rho\rbrack}\lbrack N\rbrack}\ {\mathbb{d}V}}}} & (4.10)\end{matrix}$

Where, the nodal mass matrix [ρ] includes the rotary inertia terms.

For a unidirectional lamina, the material properties matrix is given by:

$\begin{matrix}{\lbrack D\rbrack = \begin{bmatrix}\frac{E_{1}}{1 - {v_{12}v_{21}}} & \frac{v_{12}E_{2}}{1 - {v_{12}v_{21}}} & 0 \\\frac{v_{21}E_{1}}{1 - {v_{12}v_{21}}} & \frac{E_{2}}{1 - {v_{12}v_{21}}} & 0 \\0 & 0 & G_{12}\end{bmatrix}} & (4.11)\end{matrix}$

Input Variables for Sensitivity Coefficient Calculations

From the literature review, it was noted that the state variables uponwhich the response of the composite plate depends upon may be a numberof different parameters which included, the shell thickness, number oflayers and the material of the composite plate.

The studies show that the thickness of the plate, number of layers,thickness of each individual layer, stacking sequence and type ofmaterial are some of the factors influencing the impact properties.Also, some studies have been conducted with varying the impact energy byvarying impactor mass or velocity. Some other variables were studied inother researches, but the parameters that considered to effect thesensitivity coefficient are the material and geometric properties asexplained above because the output parameter “absorbed energy” isrelated to the material and geometric properties of the plate and is notdirectly related although effected by the constraint conditions and theimpactor properties and energy.

The material properties are studied in detail individually in order tounderstand the material properties which have the most profound effecton the impact behavior of the composite plate. All the materialproperties like elastic moduli in the fiber and transverse direction,shear modulus and strength under various conditions etc. are analyzedusing sensitivity analysis. The variables that are studied by thisapproach are listed in Table 21.

TABLE 21 List of Variables for Sensitivity Analysis No. VariableDescription 1 Tp Thickness of layer/ply 2 Tl Thickness of laminate 3 NNumber of layers 4 St Stacking Sequence 5 E₁₁ Elastic Modulus inLongitudinal Direction 6 E₂₂, E₃₃ Elastic Modulus in TransverseDirection 7 v₁₂, v₁₃ Poisson's Ratio in plane containing fiber 8 v₂₃Poisson's Ratio in transverse plane 9 G₁₂, G₁₃ Shear Modulus in planecontaining fiber 10 G₂₃ Shear Modulus in transverse plane 11 X_(t)Tensile strength in longitudinal direction 12 X_(c) Compressive strengthin longitudinal direction 13 Y_(t) Tensile strength in transversedirection 14 Y_(c) Compressive strength in transverse direction 15 S₁₂In-Plane Shear Strength 16 G_(f) ^(t) Fracture Toughness in longitudinaltensile direction 17 G_(f) ^(c) Fracture Toughness in longitudinalcompressive direction 18 G_(m) ^(t) Fracture Toughness in transversetensile fracture mode 19 G_(m) ^(c) Fracture Toughness in transversecompressive fracture mode

From variable 5 to variable 18, all are related to the selection ofmaterial and the sensitivity analysis is performed on these variables toget an informative guess for future material selection.

Validated Flat Plate Model as Nominal Case

The method developed in the sensitivity analysis is considering a caseto be nominal and then varying the input variables from this nominalcase by ±5%. The nominal case selected for this analysis was the same asthe one used for model validation and compared with the results ofYokoyama et al [72]. The advantage of applying sensitivity analysisusing a validated model is because of the possibility to isolate singleanswers to single perturbation of a process parameter [47].

The results and the input parameters used in the model validation wereselected as the nominal and the nominal values for the variables arelisted in the Table 22.

TABLE 22 Nominal Values for the input variables. No. Variable NominalValues 1 Tp 0.21 mm 2 Tl 4.2 mm 3 N 20 4 St[45/−45/0/90/45/−45/0/90/45/−45]_(s) 5 E₁₁ 60.8 GPa 6 E₂₂ = E₃₃ 58.25GPa 7 v₁₂ = v₁₃ 0.07 8 v₂₃ 0.4 9 G₁₂ = G₁₃ 4.55 GPa 10 G₂₃ 5 GPa 11X_(t) 621 MPa 12 X_(c) 760 MPa 13 Y_(t) 594 MPa 14 Y_(c) 707 MPa 15 S₁₂125 MPa 16 G_(f) ^(t) 160 KJ/m² 17 G_(f) ^(c) 25 KJ/m² 18 G_(m) ^(t) 10KJ/m² 19 G_(m) ^(c) 2.25 KJ/m²

The output variable for sensitivity analysis is chosen to be thedissipated impact energy or the energy absorbed during the impact event.The absorbed energy gives an account of the damage done to the compositeplate in the event of impact. The less this energy the better thedesign, considering this it is best suited for the study as improvementin the impact performance of the composite plate is sought. The impactenergy absorbed for the nominal case is 4.74 J.

Equivalent Elastic Modulus

The input variables like thickness of plate or the thickness of a singlelayer may be easily varied by 5% as defined in the approach. But, thevariables like the stacking sequence or the number of layers which arenot defined by a scalar cannot be varied in the same sense as othervariables. For the stated reason, there was a need to develop anunderstanding to vary these parameters in order to better estimate theireffects.

The material properties given in the Table 1 for the nominal case arefor a unidirectional lamina i.e. a single ply of composite materialswith all the fibers aligned in one direction. With more than one layersstacked at different orientations, they have an overall effect on thephysical properties of the whole composite plate.

The stacking sequence for the nominal case corresponds to the concept of“Quasi-Isotropic” laminate, which is the case when the equivalentmodulus of elasticity of the whole plate is same in the plane containingfibers, the other case happens to be when the modulus of elasticity inthe plane containing fibers is not equal.

The laminate stiffness matrix is given by,

$\begin{matrix}{\lbrack K\rbrack = \begin{bmatrix}A & B \\B & D\end{bmatrix}} & (4.12)\end{matrix}$

Where each A, B and D is sub-matrices defined as the ExtensionalStiffness, Coupling Stiffness and the Bending Stiffness matrices. [60]

The terms of these matrices are given by

$\begin{matrix}{A = {\sum\limits_{k = 1}^{N}\;{\overset{\_}{D}\left( {z_{k + 1} - z_{k}} \right)}}} & (4.13) \\{B = {\frac{1}{2}{\sum\limits_{k = 1}^{N}\;{\overset{\_}{D}\left( {z_{k + 1}^{2} - z_{k}^{2}} \right)}}}} & (4.14)\end{matrix}$

And

$\begin{matrix}{\begin{Bmatrix}N_{x} \\N_{y} \\N_{xy}\end{Bmatrix} = {{\begin{bmatrix}A_{11} & A_{12} & A_{16} \\\; & A_{22} & A_{26} \\{sym} & \; & A_{66}\end{bmatrix}\begin{Bmatrix}ɛ_{x}^{0} \\ɛ_{y}^{0} \\\gamma_{xy}^{0}\end{Bmatrix}} + {\begin{bmatrix}B_{11} & B_{12} & B_{16} \\\; & B_{22} & B_{26} \\{sym} & \; & B_{66}\end{bmatrix}\begin{Bmatrix}\kappa_{x} \\\kappa_{y} \\{2\kappa_{xy}}\end{Bmatrix}}}} & (4.16)\end{matrix}$

Hence,

$\begin{matrix}{D = {\frac{1}{3}{\sum\limits_{k = 1}^{N}\;{\overset{\_}{D}\left( {z_{k + 1}^{3} - z_{k}^{3}} \right)}}}} & (4.15)\end{matrix}$

For the case of Quasi-Isotropic laminates, the terms A₁₁ and A₂₂ must beequal. The nominal case selected had the quasi isotropic behavior. Inorder to study the effect of stacking sequence, it was assumed that thevariation in the overall elastic modulus should be studied. Hence, theoverall elastic modulus was considered to be varied to study the effectof stacking sequence. For the positive variation, a stacking sequencewas designed such that the elastic modulus of the laminate increases byabout 5%.

The nominal stacking sequence as listed in table 1 is[45/−45/0/90/45/−45/0/90/45/−45], had the modulus of elasticity in thelongitudinal direction is equal to be 38.7 GPa, the stacking sequencecorresponding to 5% increase in the longitudinal elastic modulus whichis 40.635 GPa is [30/−60/0/90/30/−60/0/90/30/−60]_(s). Similarly, forthe variation of −5% in the longitudinal elastic modulus which is about36.765 GPa, the stacking sequence is [60/0/45/−45/60/0/451−45/60/0]_(s).These layer configurations give the required equivalent longitudinalelastic modulus which is very close to the 5% variation.

Also, the number of layers was also selected as an input parameter,which means the variation would cause the number of layers in thelaminate to increase and decrease by 1 layer; this would result in thechange of elastic modulus of the laminate. But, it is varied in such away that the quasi-isotropic behavior of the laminate didn't change.

The change in the parameters for the sake of sensitivity analysis istabulated in Table 23 and the results of all the variables and theirsensitivity coefficient are discussed in the next section.

TABLE 23 Variation in the nominal values of the input parameters. No.Factor Units Nominal Value 5% Change X + ΔX X − ΔX X1 Tp mm 0.21 0.01050.2205 0.1995 X2 N Unitless 20 1 21 19 X3 St GPa [45/−45/0/90/ 1.935[30, −60, 0, 90, [60, 0, 45, −45, 45/−45/0/90/ 30, −60, 0, 90, 60, 0,45, −45, 45/−45]s ≈ 38.7 30, −60]s ≈ 40.7 60, 0]s ≈ 36.8 X4 E₁₁ GPa 60.83.04 63.84 57.76 X5 E₂₂ = E₃₃ GPa 58.25 2.913 61.1625 55.3375 X6 v₁₂ =v₁₃ 0.07 0.0035 0.0735 0.0665 X7 v₂₃ 0.4 0.02 0.42 0.38 X8 G₁₂ = G₁₃ GPa4.55 227.5 × 10⁻³ 4.7775 4.3225 X9 G₂₃ GPa 5   250 × 10⁻³ 5.25 4.75 X10X_(t) MPa 621 31.05 652.05 58.995 X11 X_(c) MPa 760 38 798 722 X12 Y_(t)MPa 594 29.7 623.7 564.3 X13 Y_(c) MPa 707 35.35 742.35 671.65 X14 S₁₂MPa 125 6.25 131.25 118.75 X15 G_(f) ^(t) KJ/m² 160 8 168 152 X16 G_(f)^(c) KJ/m² 25 1.25 26.25 23.75 X17 G_(m) ^(t) KJ/m² 10   500 × 10⁻³ 10.59.5 X18 G_(m) ^(c) KJ/m² 2.25 112.5 × 10⁻³ 2.3625 2.1375

The point to note here is that the values represented in Table 23, doesnot represent the realistic values of any material in terms of theelastic moduli and the strength values. Rather these values has beenadjusted according to the criteria of sensitivity analysis which statesone variable is changed at a time by a some percentage and others keepconstant and the same process is repeated for all the variables.

Results and Discussions

As per the procedure described above a total of 36 simulations wereperformed using the commercial FEA software ABAQUS to determine theamount of impact energy lost in damage during the impact process foreach of the above defined cases. The results for few of the parametervariations were as expected while there were some results that helpedunderstand the role of certain variables play in the impact behavior ofthe composite laminate.

The results for all the different cases were compiled and sorted in theorder of the calculated normalized sensitivity coefficients (NSCs). Theorder of the list provides with the information that which variable hashow much effect. The larger the NSC value, the more that variableinfluences the output variable which in this case is the amount of theabsorbed energy. The results are tabulated as shown in the table in thedescending order of NSC.

As mentioned earlier, the amount of energy absorbed in the nominal casewas 4.74 J. It is observed that based on the amount of energy absorbedin each variation of variables, the NSC has different orders ofmagnitude.

TABLE 24 provides a sorted list of parameters in descending order withrespect to the NSC. Energy Energy absorbed in absorbed in No. Symbol X +ΔX (J) X − ΔX (J) NSC X1 Tp 4.59 5.16 1.4096 X3 St 5.59 5.34 0.2899 X10X_(t) 4.66 4.87 0.2001 X2 N 4.77 4.88 0.0609 X15 G_(f) ^(t) 4.70 4.810.0479 X5 E₂₂ = E₃₃ 4.80 4.70 0.0440 X4 E₁₁ 4.77 4.72 0.0117 X6 v₁₂ =v₁₃ 4.78 4.75 0.0056 X16 G_(f) ^(c) 4.74 4.77 0.0042 X8 G₁₂ = G₁₃ 4.744.76 0.0024 X11 X_(c) 4.77 4.75 0.0015 X17 G_(m) ^(t) 4.76 4.74 0.0014X13 Y_(c) 4.77 4.75 8.22 × 10⁻⁴ X14 S₁₂ 4.74 4.75 3.89 × 10⁻⁴ X12 Y_(t)4.75 4.76 3.35 × 10⁻⁴ X18 G_(m) ^(c) 4.76 4.76 1.69 × 10⁻⁴ X7 v₂₃ 4.764.76 1.28 × 10⁻⁴ X9 G₂₃ 4.74 4.73  6.4 × 10⁻⁶

The results of the calculated NSC are graphically represented in theFIGS. 14A and 14B. In particular, FIG. 14A illustrates a graph ofnormalized sensitivity coefficients for variables demonstrating arelative effect of each absorbed impact energy. FIG. 14B illustrates agraph of normalized sensitivity coefficients for variables having agreater influence on the amount of absorbed energy except thickness. Theresults listed in Table 24 indicate that there is a large dependence ofthe impact performance of composite plates on some parameters such asthe thickness of the layer, number of layers, stacking sequence and thematerial properties like the tensile strength and the fracture toughnessin the fiber direction. The other material properties studied showeddependence of the impact performance does not vary that much with thevariation of standard 5% from the nominal values.

According the values listed in Table 24, the parameters considered tohave significant effect are:

1) Thickness of each layer/ply

2) Stacking Sequence

3) Tensile strength in the fiber direction

4) Number of layers

5) Fracture toughness in the fiber direction during tensile loading

The parameters 3 and 5 are related to the material properties and arehence dependent upon the material selection. These parameters will helpin selection of material for the fiber and matrix material.

The effect of thickness of individual layers show that the increase inthickness results in the decrease in absorbed energy and of all theparameters considered the effect of thickness is most profound on theimpact performance of the composite plate. This result is intuitive andin accordance with the available studies in the different literature.The effect of increasing thickness of individual layers has been studiedextensively and is the most effective parameter to increase the impactresistance of composite plates. This result is backed by the availableresults from the studies of Zhao et al. [73]. Zhao et al. demonstratedthat with the increasing thickness the damage is considerably reducedwhile the stacking sequence was kept constant. As discussed in theliterature, the effect of thickness is most prominent among all thevariables considered, is also supported by the fact that the value ofNSC for the case of thickness variation is the highest whichcharacterizes a strong dependence on the thickness of the layer.

The first parameter to study was the thickness of the individual layer,the effect of the variation reveals that the impact performance improvesas the thickness is increased, i.e., the amount of energyabsorbed/dissipated decreases as the thickness is increased and viceversa.

The second most important parameter is found to be the stackingsequence, one important aspect to understand is that the stackingsequence effect is not linear considering that in this study thestacking sequence is studied in terms of the equivalent elastic modulusof the whole laminate. The nominal case that was selected to be thequasi isotropic behavior has the best performance in terms of minimumimpact energy absorption.

The results show that the minimum amount of energy absorbed is for thecase where the laminate configuration is such that the laminate behavesas quasi-isotropic material. This result agrees with the result from thestudy of Aktas et al. [2]. The value of the NSC calculated for thevariation in the stacking sequence suggests that the impact resistanceof composite laminated plates is highly dependent upon the stackingsequence. The dependence is not linear and as the stacking sequenceconverges to a quasi-isotropic behavior the amount of impact energyabsorbed is significantly reduced.

The other important factors were the tensile strength and the fracturetoughness in the fiber direction, increasing these parameters result inlower impact energy absorption while lowering these values has inverseeffect. The tensile strength of the fiber is the third most significantvariable as observed by the NSC and is quite close to the NSC ofstacking sequence. This has a significant effect on the understanding inthe design process of structures with composite materials that aresusceptible to the impact loading due to low velocity impacts. It issuggested that the material should be chosen as such which offersgreater tensile strength as compared to the other material properties.

The tensile strength of the fiber has a significant effect on theabsorbed impact energy as described by the damage initiation equationsby Hashin (1980) given by equation (3.5), during the impact loading theplate is stretched and due to plate in tension as evident in FIG. 10A,the tensile strength of fiber plays an important role in the impactbehavior of the composite plates. As evident by equation (3.5), thehigher the strength value, the more stress it may bear before breakagehence less absorbed energy and better impact resistance.

Similarly, the effect of fracture toughness may be observed from thedamage evolution laws described by Hashin. As shown in FIG. 6, theamount of dissipated energy is the area under the curve for theequivalent stress-displacement curve, the higher the fracture toughnessthe more stress composite plate may withstand before the damage.

Finally, the last factor considered was the number of layers; itsimilarly has not a linear relation like the stacking sequence.Increasing and decreasing by 5% the number of layers while keeping thetotal laminate thickness constant result in increased impact energyabsorption. Hence, it may be deduced that there must be an optimalnumber of layers for a fixed thickness which will give the better impactperformance. This observation may be related to the fact that fromvarious studies it is observed that increasing layers with 90°orientation results in the increase in contact force as described byTiberkak et al. [65]. Thus, there must be an optimum condition for whichthe amount of absorbed energy and the resulting damage will be minimum.

The other material properties that have slight influence on the impactresistance are the fracture toughness of the material in the tensileloading in the longitudinal direction and the transverse elasticmodulus. Both these variables have the NSC values in the same order asthe NSC for the number of layers but slightly less. Besides, both thesevariables having similar NSC values, their behavior is completelydifferent. The increase in the fracture toughness G_(f) ^(t) results inbetter impact performance while the increase in transverse elasticmodulus results in the increase in impact energy absorption hence it isdesirable to have transverse elastic modulus low.

The rest of the material properties like the longitudinal elasticmodulus, the shear modulus and the strength of the lamina in thetransverse direction have very small effect on the overall impactperformance of the composite plate. This may be observed by the factthat the value of NSC is of one or more order less than the NSC ofstacking sequence and number of layers etc.

This section presented the model validation and the sensitivity analysisapproach to ascertain the effects of various geometrical and materialproperties of composite materials on the impact performance of thecomposite laminated plates. Initial numerical model was selected fromthe literature and the results verified against the available numericaland experimental results. The results show quite a good agreement withthe experimental results. The model was then selected as the nominalcase for further evaluation of NSC using the commercial finite elementsolver ABAQUS explicit.

The results presented in the current study gives an insight about theeffects of the considered parameters on the impact performance in termsof a normalized coefficient. The advantage of such a coefficient is thatan equal amount of variation in any of the parameters will behighlighted in varying effect on the output; hence, it may be classifiedaccording to the order. ABAQUS explicit solver was used to perform thefinite element simulations to find the effect of variations in all theinput variables one at a time on the absorbed impact energy. The amountof energy absorbed varies significantly for the variations in thethickness of a single layer, number of layers, stacking sequence and thematerial properties that have significant effect were the tensilestrength of the layer in the fiber direction, fracture toughness of thelaminate in the tensile loading in the longitudinal direction and tosome extent the elastic modulus of the transverse direction has effecton the absorbed impact energy. The only peculiar behavior is of thestacking sequence, as the stacking sequence is changed the overallelastic modulus of the laminate varies and as the behavior of thelaminate moves away from that of the quasi-isotropic the amount ofenergy absorbed increased resulting in a poor performance compared tothe nominal case.

The results from this study will help the authors in the future work indesigning composite laminated plates having better impact resistance.The results will allow a more methodical approach in selecting theparameters to vary in order to achieve better impact performance ofcomposite laminates against the low velocity impact loadings. Thus, theresults from this section for the improvement of impact performance ofcomposite plates may be summarized as follows:

The layer thickness has the most prominent effect with the more thethickness, the better the impact resistance.

Stacking sequence should be such that the overall behavior of thelaminate should be close to quasi-isotropic.

The most important material property for selection of material is thetensile strength of the fiber in the longitudinal direction.

The number of layers has an effect on the impact resistance and shouldbe selected carefully as to not just increase the layers which result inmore contact force and hence greater damage.

Parametric Study of Design Variables

The main idea of this current study was to investigate the relation ofthe amount of damage occurring during an impact load with the number offactors such as the thickness of the layers, number of layers,orientation angles, material types and inclusion of other materials.These factors were identified using the sensitivity analysis approachdiscussed in the previous chapter. This way a more knowledgeable designcriterion may be developed which will be optimal in terms of performanceand the cost of the material. For a comprehensive study of effects ofthese factors, a design of experiments approach was adopted where allthe possible combinations may be tried.

In this section, design of experiments approach is presented along withthe discussion of the effects of these factors on the impact performanceof both the composite plates and the pipes.

Design of Experiments

Design of experiments is a very useful tool to investigate the causesand effects of various factors spread over a domain. The use of designof experiments along with the finite element analysis gives an analyst apowerful tool to understand deeply the variations of the outcomes of aprocess and the factors causing these variations. In addition to theeffects on the design or process, design of experiments givesstatistical significance to understand them. The combination of finiteelement analysis with the design of experiments provides the opportunityfor the current work to study the complete domain of the identifiedvariables from sensitivity analysis and their relationship with theimpact performance. It is evident from the literature review that so farthe experimental studies conducted in the low-velocity impacts on thecomposite materials have not been comprehensive. This is due to theobvious reasons that the production of such large number of samples iscostly and the experiments for impact loads may be classified asdestructive analysis. Therefore, most of the studies conductedexperimentally considered few variations in the factors like thicknessor the stacking sequence.

The use of finite element analysis is therefore beneficial andadvantageous to combine with the large number of experiments designedusing DOE.

Numerical Experiments for Flat Plate

Previously discussed, the sensitivity analysis characterized fourvariables namely the thickness of the single layer, number of layers,stacking sequence and the material type to be of most significanceconsidering the impact behavior. Therefore, here these four factors areconsidered in the DOE study and the different levels studied are listedin the Tables 25 and 26. Here, the materials considered are onlycarbon/epoxy and the glass/epoxy and the tables are listed separatelyfor both these materials. This is due to the fact that the carbon beingthe stronger material has different thickness ranges in which it variesfrom completely damaged, i.e., penetration of the striker to thecomplete survival, i.e., the striker bounces back with the same speed.The stacking sequence is kept the same for both of these materials asthe effect of stacking sequence in both the materials had to becompared.

An initial DOE was fashioned with three discrete levels of thickness andfour discrete levels for the number of layers. These were from levels‘5’ to ‘7’ for the thickness and levels from ‘1’ to ‘4’ for the numberof layers. This combination of factors results in total 96 experimentsfor both types of the materials. But after the initial simulations itoccurred that there are two shortcomings in this design. One, in thisrange the variation was not from complete damage to complete survival;it only showed the intermediate behavior. Two, the number of experimentswere not sufficient enough for a good training of neural networks.Therefore, additional levels were added for both carbon and glass fiberplates to observe the complete spectrum. The results were calculated interms of the absorbed energy with the impact energy fixed at 27.55 J.The impactor dimensions, weight and velocity are being kept constant inall the cases. The boundary conditions are also kept the same throughoutall the experiments. The simulations were performed in ABAQUS Explicitenvironment.

TABLE 25 DOE Table for Carbon/Epoxy Composite Plates. Thickness (mm)Number of Layers Stacking Sequence 0.12 16 [0/30/60/90] 0.14 20[45/−45/0/90] 0.16 24 [45/30/−30/−45] 0.18 28 [60/45/−45/−60] 0.2 320.25 36 0.3 0.35 0.4

In total, 108 experiments were performed using the carbon/epoxy as thematerial for the plate. The results are quite large and are listed inthe Appendix A.

TABLE 26 DOE Table for Glass/Epoxy Composite Plates. Thickness (mm)Number of Layers Stacking Sequence 0.25 24 [0/30/60/90] 0.3 28[45/−45/0/90] 0.35 32 [45/30/−30/−45] 0.4 36 [60/45/−45/−60] 0.45 0.50.6

Similar to the experiments conducted numerically for the carbon/epoxyplates, 108 experiments were performed for the glass/epoxy plates aswell. The combinations were not all similar but the initial 48experiments were kept. All the results are listed in the Appendix A fromTable A. 1 to Table A. 8.

Effects of Fiber Material

In this study, only two materials have been selected for the comparison.The material properties are listed in the Tables 3, 5 and 7 for thecarbon/epoxy composite plate and for the glass/epoxy in the Tables 4, 6and 8. The simulations were designed such that a direct comparison maybe obtained between the absorbed energy by the two materials. Theelastic moduli of the carbon fiber system are greater than the glassfiber system. Also, the difference in the strength levels is alsoconsiderably high in favor of carbon based composites as well as theintralaminar fracture toughness values. The damage mechanism mentionedin the equations (3.3) to (3.6) is based upon the strength levels of thecomposite. Once, the damage is initiated the cracks propagate throughthe material which is modeled using the linear energy based damageevolution model. According to the material properties, the carbon/epoxysystem should be better than the glass/epoxy system in terms of impactperformance as both the strength and the fracture energies forcarbon/epoxy is higher.

The results from the simulation were intuitive as the carbon/epoxycomposite plate has better impact resistance compared to glass/epoxycomposite plate at the same conditions of thickness, stacking sequenceand boundary conditions.

FIGS. 15A-15D illustrate that the composite plates of CFRP are betteragainst impact loads compared with the GFRP. This has already beenexplained above is due to the higher strength and the fracture energiesof the carbon/epoxy. The results listed in the tables in Appendix A forthe composite plates demonstrates that the composite plates failcompletely at the thickness level of less than 6 mm for GFRP while thatof CFRP may withstand the same impact load at around 2 mm.

Effects of Thickness of Plate

The sensitivity analysis showed that the biggest single factor in theimpact performance of composite structures is the overall thickness ofthe plate or the pipe. In this section, the effect of thickness of bothCFRP and GFRP plates are discussed.

From the results of sensitivity analysis, it is observed that theincreasing thickness reduces the amount of absorbed energy.

A) Carbon/Epoxy Plates

This trend may be observed in the graphs for the various thicknesses forcarbon/epoxy plates shown in the FIGS. 16A-16D.

The results from the ABAQUS analysis follow the intuition that with theincreasing thickness the amount of absorbed energy will decrease i.e. animprovement in the impact performance of the composite plate. Theresults as plotted against the overall thickness of the plate show thatthey follow a certain trend and as a basic trial, a simple fourth orderpolynomial was fitted over the scattered data. The curve approximatesquite accurately except for a few results which were away from the trendline.

FIG. 17 illustrates a force vs. time plot of CFRP plates of twodifferent thicknesses using the [0/30/60/90] laminate configuration.

B) Glass/Epoxy Plates

It was expected that the composite plates with glass fiber as thereinforcement material will behave in a similar fashion as the carbonfiber based plates did. However, it was found that the behavior of theglass/epoxy plates was a little peculiar as initially with the increasein thickness the absorbed energy reduced, therefore, improving theperformance of the composite plates against the low velocity impactloads. However, a further increase in the overall thickness of the plateeither by means of increase in layer thickness or by increasing thenumber of layers resulted in the decrease in performance. This behavioris strange and as compared with carbon/epoxy system was not observed inthose cases. This behavior is clearly seen for all the different casesof stacking sequence. But, there is a thickness value beyond which theimpact resistance starts to increase again and eventually the platealthough at very large thickness performed without significant damage.These results are presented in the graphical form in the FIGS. 18A-18D.

Physically, this phenomenon may be explained such that when thethickness of the composite plate is small, the plate behaves much morelike a membrane and during impact the plate stretches until all thekinetic energy is transferred to the plate and then it pushes back theimpactor giving away some of the energy back to the impactor and therest is dissipated in the form of damage within the plate. The more thethickness of the plate is increased, the stiffer it gets and the abilityto bend under impact loads is reduced which increases the bending stressand hence the plate suffers more damage. At very high thickness, theplate becomes very strong and stiff which results in very low amounts ofenergy absorbed.

This large increase in the absorbed energy at the intermediate thicknessrange may be explained by the concept that the flat plates with smallthickness acts like a membrane and in such thin plates the compressionfailure in the plane of the plate or through the thickness cannot beobserved. In such cases, the maximum deformation is higher than theplate thickness [26]. This may be observed in this case as well, forexample, if one considers the cases number 1 and 2 from the Table A. 5,the plate with thickness 6 mm absorbs less energy than the plate with7.2 mm. The maximum deformation in the case 1 here was found to be 8.7mm which is more than the plate thickness while the maximum deformationis about 6.5 mm in the case of plate with 7.2 thickness. Anotherinteresting point observed was the calculation of the A, B and Dmatrices defined in the equations (4.12) to (4.16). It was observed thatirrespective of the stacking sequence, the increase in absorbed energywas coincident with the same values of the sum of the members of theextensional stiffness matrix A. It is to be noted that the extensionalstiffness matrix provides the relationship between the strains and theforces for the laminate. This value was quantified to be in the range of330 to 450 GPa-m.

The same effect may be observed in the carbon/epoxy plates but becausethe fracture energies are high for carbon, the increase in absorbedenergy with the increase in thickness is not high. Although, in somecases in FIGS. 16A-16D, where the amount of energy to increase slightlyor at least didn't decrease by the same percentage as was expected.

FIG. 19A illustrates a force vs. time plot of GFRP plates of twodifferent thicknesses using [45/−45/0/90] laminate configuration.

A comparison of force history graphs for two carbon/epoxy laminates andtwo glass/epoxy laminates show that in the glass/epoxy plates the sharpfalls in the force. A sudden fall in the force represents the onset ofdamage until the impact load is supported by the layers so far remainundamaged. In carbon/epoxy plates, the fall in force is not that highbefore it starts to increase which shows that after the initiation ofdamage it doesn't propagate so quickly. This is due to the high fractureenergies of the carbon fiber. Whereas the glass/epoxy plates show a muchsteeper fall in the force values. This represents that the damage onceinitiated may propagate quite easily, that is due to the low fractureenergies of the glass fiber. A comparison of the fracture energies showsthat glass has 1/16^(th) of the fracture energies of the carbon. Tobetter understand this cause, a similar glass/epoxy plate as shown inFIG. 19B was simulated with 4 times the initial fracture energy only inthe tensile fiber direction. The phenomenon that impact performancedeteriorates in these samples with the larger thickness was stillobserved but this time the difference in the amount of absorbed energiesis quite low and also the force history graphs are close to the one ofcarbon/epoxy.

Effects of Stacking Sequence

The stacking sequence of composite laminas is the arrangement of theindividual layers in specific orientation. The ability of arranginglayers according to the design gives the special advantage to compositematerials over the conventional isotropic materials in better loadhandling capabilities. In this study, four different stacking sequenceswere studied. These stacking sequences are arrangement of 4 layersmentioned in Tables 25 and 26 in various directions which were thenrepeated to achieve the desired thickness and number of layers.

FIGS. 20A and 20B illustrate absorbed energy vs. thickness for stackingsequences 1-4 for carbon/epoxy systems. In particular, FIG. 20A plotsthe amount of absorbed energy for all the stacking sequences studied forthis work. FIG. 20B illustrates a comparison of absorbed energy forstacking sequences for thin CFRP plates.

From these plots, it is observed that for the thinnest plates, the worststacking sequence was 4 while the best was stacking sequence 1. This isin-line with the current research where the quasi-isotropic behavior oflaminate configuration of stacking sequence 1 is suggested to be thebest against the low-velocity impacts. This suggestion is correctconsidering the stacking sequence 1 laminate configuration distributesfibers equally in both the principal directions which are the main loadbearing component in the composite materials and have equal stiffnessesin the x and y directions. The other two stacking sequences lie inbetween the stacking sequences of 2 and 4.

Further increasing the thickness provides more insight into the effectsof stacking sequence. Here, it may be observed that in the intermediatethickness range for the carbon/epoxy plates, the best laminateconfiguration or the stacking sequence is the sequence number 4.However, at larger thicknesses, there is not much of a difference. FIG.20C shows the similar behavior for the glass/epoxy plates

FIG. 20D illustrates a comparison of absorbed energy for stackingsequences for thick CFRP plates.

Even for the glass/epoxy plates the stacking sequence 4 is better interms of the absorbed energy for the moderately thick plates. This isdue to the fact that the most important factor in the damage limitationis having a high tensile strength in the fiber directions. Therefore, toavoid damage due to the impact loads, the fibers should be aligned inthe direction where the maximum stress is observed. If you recall theboundary conditions that were applied for the numerical model as shownin FIG. 7A, it was on the two shorter edges of the plate which makes theplate constrained in the global y-axis direction. The reason that thestacking sequence 4 has better performance is down to this reason, sinceit has more fibers aligned towards the y-axis and it may withstand moreloads in this direction. A simple calculation of the transformation ofthe strengths in the x and y-axis direction show that the stackingsequence 4 has the highest strength in the y-direction followed bysequence 2. Therefore, this stacking sequence offers better performanceespecially in the intermediate thickness plates where the others werefacing more damage.

Alternative explanations for this kind of behavior may be find in thestudy by Zhao et al. [73], where they observed that the maximum damagesize and the maximum deflection of the composite plate decreases withthe increase in the bending stiffness of transverse direction. Thebending stiffness may be calculated using the equations provided earlierfor the A, B and D matrices, where D matrix represents the bendingstiffness matrix. Zhao et al. reported that the best stacking sequencewould be the one in which the longitudinal and transverse stiffness areequal. This is true in the case where one has similar boundaryconditions on all the edges. If, one has boundary conditions differenton different directions, then one may optimize the stacking sequence asis in this case the sequence 4 provides the best solution according tothe given boundary conditions.

Effects of Layer Thickness and Number of Layers

Sensitivity analysis results suggested that the number of layers havesome effect on the impact performance of the composite plates. From thesensitivity analysis, it was observed that if the overall thickness ofthe plate is remained constant but the numbers of layers vary, then theamount of energy absorbed will be varied. The results as listed in theTable A. 1 to Table A. 8 suggest that this is indeed the case. But, thevariation is not always as initially observed from the sensitivityanalysis. The variation in the amount of absorbed energy depends uponthe orientation of individual layers that are added to or removed fromthe stacking in compensation to keep the thickness constant.

If one considers the two cases from the carbon/epoxy plates with thesame thickness but different number of layers, one observes someinteresting results. It may be observed that the in case where one has20 layers, there are four additional layers of 0° and 30° two each. Asdiscussed earlier during the effects of stacking sequence, these fouradditional layers are just keeping the overall thickness constant but infact are reducing the number of fibers from the layers of 60° and 90°.As you by now know that these layers when transformed along theprincipal directions share larger share of the strength in the y-axiswhere it was deduced that the majority of the stress would be produced.Hence, the observation in this case is that the additions of these 0°and 30° layers are doing more harm than good. As may be seen in from theamount of energy absorbed increased in the case of 20 layers compared tothe case of 16 layers.

FIG. 21A illustrates a chart that compares an amount of absorbed energyfor CFRP plates with 16 layers and CFRP plates with 20 layers.

TABLE 27 Orientation of individual layers for two cases of carbon/epoxyplate Thickness of Number of Total Thickness Absorbed layer (mm) Layers(mm) Energy (J) 0.16 20 3.2 8.227648 0.2 16 3.2 7.894729 0.25 16 46.78549 0.2 20 4 7.570117

By virtue of the above explanation, it might be argued that if oneincreases the number of layers keeping the thickness constant in such away that some of the fibers from the 0° and 30° layers are removed andadded to 60° and 90° orientated layers, then one might observeimprovement in the impact performance. By comparing the results fromTable 28 and FIG. 21B, it is observed that is indeed the case.

TABLE 28 Orientation of individual layers for two cases of carbon/epoxyplate. Thickness of layer Number of Total Thickness Absorbed Energy (mm)Layers (mm) (J) 0.3 20 6 5.915478 0.25 24 6 5.849285 0.35 32 11.2 0.20310.4 28 11.2 0.56125

The same behavior may be observed for the glass/epoxy plates asdemonstrated from the results in Table 29 and FIG. 21C for the situationwhere the addition of layers results in the improvement of impactperformance. This is due to the same reason as explained above that thisis due to the addition of layers which may bear more load in thedirection of the stress and hence improve the overall impactperformance. While Table 30 and FIG. 21D represents the case where theaddition of layers decrease performance.

TABLE 29 Orientation angles of GFRP plates where increasing layersincrease performance. Thickness of layer Number of Total Thickness (mm)Layers (mm) Absorbed Energy (J) 0.35 24 8.4 14.20262381 0.3 28 8.413.02287113 0.3 32 9.6 15.32878983 0.4 24 9.6 15.3788138

TABLE 30 Orientation angles of GFRP plates where increasing layersdecrease performance. Thickness of layer Number of Total Thickness (mm)Layers (mm) Absorbed Energy (J) 0.3 36 10.8 10.72389162 0.45 24 10.810.4629225 0.35 36 12.6 8.650227096 0.45 28 12.6 8.588431527

Numerical Experiments for Composite Pipes

Following on from the study of effects of the parameters on the impactperformance of the composite plates, the study is carried out for thecomposite pipes as well. The factors that have influence on the impactperformance are the same as found from the sensitivity analysis studyand one also studied their effects for the composite plates as well. Thedifference between plates and the pipes is in the type of lamina and thelamina's orientation angle. Since, it is known that the composite pipesare manufactured using the filament winding technique, the type oflamina considered in this study is the unidirectional lamina. Thematerial properties for carbon/epoxy and glass/epoxy are listed in theTables 9, 11 and 13 and Tables 10, 12 and 14 respectively. Also, thepipes manufactured using filament winding technology have only twoorientations, i.e., ±0, therefore here it was studied different windingangles rather than a combination of layer orientations as studied forthe composite flat plates.

The design of experiments is again applied to gather the results for thecomposite pipes where the complete damage to the complete survivalconfigurations is selected. The layer thickness is selected based on thecommercial availability of carbon and glass fibers. The winding anglewas selected from 35° to 75° with an interval of 10°. This is selectedon the basis of the study of Rosenow [59] in which he studied the effectof variation of winding angles on the filament wound glass fiberreinforced polyester. The author studied winding angles from 15° to 85°.In his study, the author suggested that the winding angle of 55° wasoptimal for the hoop to axial stress ratio of 2, while for only pressureloadings without axial stress the optimal winding angle was 75°.

Initially, experiments were designed with 4 distinct layer thicknesses,5 sets of number of layers and 5 different winding angles. This designgave a total of 100 simulations to be carried out. Later on, twoadditional layer thicknesses and a further set of simulations were runwith total number of layers up to 40 for the glass fibers and 16 layersfor the carbon fibers. These simulations were added for the reason tohave all the variation from maximum damage to minimum damage. Also,after the initial simulations it was noted that more simulations shouldbe tried around the mean angle of 55° and therefore 4 new winding angleswere added. Table 31 and 32 give the values for all the selected factorsfor glass/epoxy and carbon/epoxy pipes respectively.

TABLE 31 DOE table for the GFRP pipes. Thickness Number of LayersWinding Angles 0.25 20 35 0.3 24 45 0.35 28 50 0.375 32 52.5 0.4 36 550.425 40 57.5 60 65 75

TABLE 32 DOE table for the CFRP pipes. Thickness Number of LayersWinding Angles 0.25 16 35 0.3 20 45 0.35 24 50 0.375 28 52.5 0.4 32 550.425 36 57.5 60 65 75

The load and the boundary conditions are applied as described in thenumerical model earlier. The impact energy of 40 J was applied using aninitial velocity given to the striker. In total, 162 differentcombinations were simulated for each glass and carbon fiber basedcomposite pipes. The results are tabulated in the appendix A from TableA. 15 to Table A. 30. In total, 162 simulations were performed for eachcarbon and glass based composite pipes.

Effects of Fiber Material

Two types of materials carbon and glass are used as fiber reinforcementfor the composite pipes. It is clear from the material properties Tables9 to 14 presented earlier that the carbon fiber is much stronger thanthe glass fiber and also the fracture energies are higher for thecarbon. For this reason, it may be argued that the CFRP pipes willperform better under impact loads than GFRP and indeed this is the caseif one looks at the results presented in the Table A. 15 to Table A. 30.For the same geometric conditions, the amount of absorbed energy in theCFRP pipes is lower than that of GFRP pipes.

From the scatter plots of the two types of material presented in FIGS.22A-22E, the observation that the carbon/epoxy pipes will perform betterthan the glass/epoxy pipes is correct. From these plots, it may also beobserved that the difference in the amount of damage, which is theabsorbed energy, is very high in thin walled pipes. But, in themoderately thick walled pipes, although the carbon pipes perform betterbut the difference is reduced. But, in very thick pipes, again thedifference becomes quite significant.

Effects of Thickness

The increase in the overall wall thickness of pipe is assumed to besignificant by virtue of the results previously observed in thesensitivity analysis approach and also the results from the analysis offlat plates demonstrated the same phenomenon. FIGS. 23A-23J illustratethe variation in the amount of absorbed energies for both the GFRP andCFRP pipes vs. thickness. It may be observed from these plots that thereis a range of thickness values for all the winding angles and for bothmaterial types, during which the increasing thickness doesn't improvethe impact performance. The same phenomenon was observed by Zhao et al.[73] in their study, they noticed that the damage threshold velocity wasnot affected by the increase in the thickness of the plates. Theyfurther reported that although the damage is almost unaffected but thedamaged area reduced with the increase in the thickness of the plates.The same results may be deduced for this range of thickness values wherethere is no improvement in the impact resistance of the pipes. But, onedoes observe that after crossing this thickness range a sudden drop inthe amount of absorbed energy which is not reported in the study of Zhaoet al. as they studied only three cases for the thickness variations.

In all of these cases where the increase in thickness doesn't improvethe impact resistance, it was observed that the vibration in the pipeincreased. This as explained earlier for the case of flat plates, whereit was observed an increase in the amount of absorbed energy is due tothe fact that the overall stiffness of the structure increases withincrease in the thickness but not sufficient enough. Hence, one observesvibrations which is the cause of more absorbed energy as explained byKrishnamurthy et al. [36] in their research that the energy absorbedupon impact is the sum of the strain energy and the kinetic energy ofeach of the modes of vibration.

Effects of Winding Angle

The choice of the variation of the winding angle depended upon the studyby Rosenow [59]. Based on the conclusions provided by Rosenow, 5different winding angles from 35° to 75° with an interval of 10° wereselected. The results show that the pipes with winding angles of 35° and75° are particularly worse than the rest in handling the impact loads.The results are represented in the FIGS. 24A and 24B it is observed thatin most of the cases the pipes with winding angle of 55° have the leastabsorbed energy and better impact resistance.

From the above graphs, it is clear that for most of the pipe thicknessirrespective of the fiber material, the orientation of 55° performsbetter. In some cases, 45° and 65° winding angles were slightly better.In order to further examine, a further cases were simulated with anglesranging from 50°, 52.5°, 57.5° and 60°.

Close observation of these figures and the relevant tables in theappendix show that for glass fiber pipes, there is not much of adifference in terms of absorbed energy for smaller thickness pipes.However, when the thickness is increased the 55° winding angle pipeswere better in performance compared to the rest. The observation isreversed for the carbon based pipes where at smaller thickness 55° wereslightly better and increasing thickness results in slightly worseperformance but the difference is not that much.

FIGS. 25A and 25 B illustrate variations in absorbed energy with respectto winding angle for CFRP pipes and GFRP pipes, respectively.

The reason for the better performance of 55° winding angle pipes is thatthe impact force tries to bend the pipe near the impact point, hencewinding angles of 55° which is reported to perform better when loadedwith both hoop stress and axial stress performs better in the case ofimpact loads as well. The slight difference may be attributed to otherreasons such as the number of layers, thickness of the wall or theboundary conditions.

The fiber orientation in the 55° winding angles is better in terms ofperformance because it may carry both axial and hoop stresseseffectively as stated earlier and during impact, the bending due to theloads creates stress in both the longitudinal and circumferentialdirections. Hence, 55° winding angle is preferred. If one bases furtheranalysis on this assumption, it may be inferred that the winding anglesof 35° or even less are since more aligned with the longitudinal axis ofthe pipe may withstand more longitudinal or axial stresses but will beweaker in the circumferential direction. On the other hand 75° windingangles are more close to the hoop winding which is known to handleinternal pressure and hoop stresses will perform better in the loadingsthat put the pipe under circumferential stresses but will fail in theaxial loadings.

Effects of Layer Thickness and Number of Layers

It is reported in the work of Zhao et al. [73] that the stackingsequence has some major influence on the impact performance of thecurved shells. It is reported in their work that the damage is reducedwith the increase in the interface number in the laminated shells. Theinterface number may be understood as the number of time the fiberorientations are changed within a laminate. For example, in a laminatewhere [45/−45/0/90] is the stacking of layers, there are 4 interfacechanges while in a laminate where [45₂/−45₂/0₂/90₂] is the stacking oflayers the interface number is still 4 in spite of the fact that thenumber of layers are twice that of the earlier sample.

The above result suggests that in the pipes where the individual layershave less thickness and to achieve the overall thickness of the pipenumbers of layers are increased will be better than the other way round.The simulations performed however, suggested that the increase in numberof layers keeping the overall thickness constant has little effect onthe amount of absorbed energy or in some cases it has adverse effect.This may be observed in the graphs of FIGS. 26A-26J. Here, the resultsare not always in the favor of more layers. This may be due to a factthat the results described in the study of Zhao et al. [73] is forcurved shell and the current study is for pipes, therefore, the geometryand the boundary changes may affect the influence of number of layersdifferently than for flat plates or curved plates.

Inclusion of Embedding

From the study so far, it is understood that the tensile strength of thefiber and the fracture energies of the fiber materials are one of theimportant contributors. In order to improve the performance of thecomposite plates and pipes, it is therefore advisable to use materialssuch as carbon or graphite which have higher tensile strengths andfracture energies. But, the cost factor is also important since the ideais to design such that the performance is optimal with respect to theminimum costs.

To achieve better impact resistance at a lower cost is the main aim.Because the glass/epoxy systems are less expensive compared tocarbon/epoxy. This section discusses the kinds of materials included andtheir placement in the glass/epoxy plates and pipes to enhance theoverall impact performance of the structures.

Embedding Type

For the composite flat plates, the main fiber material is chosen to beglass with addition of carbon fibers. Since, the flat plates aremanufactured using the woven fabric only the carbon/epoxy woven fabricwas used along with the glass/epoxy fabric. From the material propertiesTables 1-14, it is known that the carbon/epoxy laminas are much strongerthan the glass/epoxy. Also, from the simulations run for both type ofmaterials and the results listed in the tables in the Appendix Aconfirmed that the carbon/epoxy plates perform much better than theglass/epoxy. Therefore, some layers from the composite plate werereplaced by carbon fibers. The studies prior to this one alreadyconcluded that for the case considered in this thesis for compositeplates, the best stacking sequence will be number 4, i.e.,[60/45/−45/−60], which were used in order to study the effects ofinclusion of other fiber materials. The other conditions of the load andthe boundary conditions and the impactor remain the same as in the studyfor the composite plates.

Similarly, for pipes the results already studied were utilized toenhance the impact resistance of the composite pipes made using glassfiber filament winding by the addition of other materials in the windingprocess. Usually, the pipes are manufactured using continuous filamentwinding of one type of fiber material impregnated with the epoxy resinbut it is not impossible to break the fibers after completion of layersand then include other fibers with the same epoxy resin to improve theperformance. In fact, it is a common practice in the aerospace industryto manufacture composite rocket motor casings with different windingangles and different kinds of fibers to achieve the desired designcriteria which is mostly dependent upon multi-loads situation to beencountered during service. For the composite pipes, it was observedthat the pipes with carbon fiber offer quite an advantage over the pipeswith glass fiber. But for both the types of material the best windingangle was the same as 55°. Also, inclusion of a layer of woven fabricmay be studied as it may be beneficial considering the woven fabric hasbetter strength characteristic in both the directions compared to theunidirectional lamina. Therefore, for this study inclusion ofunidirectional carbon/epoxy and woven carbon/epoxy layers in theglass/epoxy composite pipes have been studied. For the study, the loadsand the boundary conditions are kept the same as in the previous studiesdiscussed above.

Effects of Placement of Embedding

The inclusion of other materials alone cannot guarantee an increase inthe performance of the structure, the placement of the embedding is alsonecessary. Since, the material to be included is based on the superiorstrength and better performance, it should be placed where the damageinitiates. To understand the relation between the placement of theinclusions and the impact performance, different placements were triedfor the carbon layers in the composite plate that mainly consisted ofglass fibers. Following different combinations were tried with theposition of woven carbon lamina as:

Top and Bottom layers

Middle 2 layers

Top 2 layers

Bottom 2 layers

Single top and bottom layers and 2 middle layers

Single Top and Middle layers

In addition to different positions, different thickness of the carbonlayers and glass layers were considered. As described earlier, thestacking sequence considered is number 4 i.e., [60/45/−45/−60].

The results for these simulations are tabulated in the Appendix A. TableA. 9 to Table A. 14 and the graphical representation is provided in theFIGS. 27A-27C. From the results, it is evident that the greatest effectof the inclusion and placement of the carbon layers is when the overallplate thickness is small. Once, the plate thickness is increased, mostof the load bearing capacity is taken by the glass fiber layers andhence effectiveness is not measured. In the low thickness plates, theplacement of the carbon fiber layers is thus important and it isobserved that the most efficient placement when one carbon layer isplaced at the top and one in the middle. The top 2 layers of carbonperform slightly worse but this may be attributed to the fact that thefirst layer is 60° while the second one was only 45° compared to thecase where one top and one middle layers are replaced both of them beingthe 60° layers. Another important result to be noticed that the increasein the absorbed energy observed for the glass/epoxy systems FIG. 18Dwith the increase in thickness was negated quite a bit by theintroduction of carbon/epoxy layers especially when these layers arereplaced at the top only, top and middle and cases with top and bottom.Therefore, it may be deduced that the carbon layers introduction at thetop and middle gives the better impact performance at a slightly highercost.

Similar procedure was adopted to study the effect of carbon/epoxylayers, both woven and unidirectional layers, on the impact performanceof the composite pipes. From the results presented earlier, it isinferred that the damage initiates at the top layer that is the closestlayer to the impact point. Therefore, different layer combinations withwoven fabric and unidirectional fibers were tried and the results arepresented in the Appendix A. Table A. 31 to Table A. 34. The windingangle was kept at 55° as it was found out to be the optimal angle ofwinding against the impact loads.

The results suggest that the inclusion of top layers as the woven carbonfabric doesn't improve the impact performance. This is due to the reasonthat the most important strength factor in withstanding impact loads isthe tensile strength and in this case the tensile strength ofunidirectional glass fiber is slightly more than the woven carbonfabric. The inclusion of woven carbon fabric is thus not recommended asit will increase the costs without increasing the impact performance. Onthe other hand, the inclusion of unidirectional carbon layers suggeststhat there is an advantage especially when top 4 layers were replaced.In the case of top 4 layers of carbon fibers, the impact performance isin fact better than the pure carbon based pipes. This is thereforehighly recommended configuration considering less expensive with betterresistance against impact loads. The results are graphically representedin FIGS. 28A and 28B.

This section includes the results and discussion for the simulationscarried out in order to study the effects of various parameters upon theimpact performance of the composite structures. These parameters wereidentified by the sensitivity analysis but their exact nature and theexplanation of their behavior cannot be provided by the sensitivityanalysis. The approach considered in this chapter was to design a set ofexperiments to be performed numerically. Simulations were performedusing ABAQUS explicit for both flat plates and pipes. The designvariables and their effects have been studied in detail. Few of theseparameters have already been studied in the available literature and theresults from the current work is studied and compared with the alreadyavailable literature. The parameters studied in this chapter wereselected after the sensitivity analysis and were selected such that theyare directly related to the designing of the composite plates and pipes.Factors such as the impactor mass, geometry and the boundary conditionswere kept constant as most of the times in real life applications thesefactors will be outside the control of the designer. The simulationswere performed in two phases initially a complete DOE table wasconstructed but later on more variations of the factors were added tocomplete the analysis in a way that the complete range of variables isselected from being safe to complete penetration of the impactor. Themain conclusions drawn from this chapter are summarized as follows:

The most profound effect of all the variables was that of the thicknessof the plates and the pipes. The crucial observation in the analysis ofthis factor is that the dependence of impact performance on thethickness of the structure is not directly proportional. In fact, it wasfound that there was a range of thickness where actually the performanceis worse than before. This observation is explained by the ability ofthe thin structures to withstand bending without undergoing vibrations.The increase in thickness increases the structural rigidity which inturn effects adversely due to the unnecessary induced vibrations uponimpact.

The stacking sequence of the composite plates has a significant role inthe impact performance. Although not directly studied, this is due tothe boundary conditions effect. It is suggested that during the designphase knowledge of the kind of boundary conditions is better. Hence, itis recommended that the more fiber should be aligned in the directionwhere the boundary conditions are such that they restraint the bendingof the plate.

The conclusion from the chapter 4 that the numbers of layers have aneffect but they have to be chosen carefully is further explained basedon the orientation of the added layers. It is important to have as moreas possible fibers in the direction of the maximum stress during theimpact to delay the damage initiation process.

The material properties which have a significant effect on the impactresistance of any composite structures are the tensile strength of thefiber and the energy release rates during damage propagation. Care mustbe taken in the selection of material and designing of the compositestructures as to maximum utilize the tensile strength of the fibers.

The improvement in the impact resistance of the composite plates withoutincreasing costs by much may be achieved through the introduction ofcarbon/epoxy layers in place of glass/epoxy layers. These layers ofcarbon/epoxy should be introduced in places where the damage initiates.Also, the layers to be replaced should be selected carefully keeping inmind that those layers should be replaced that increases the bendingstiffness of the plate.

The best stacking sequence or the orientation angles of the layers isthe one that aligns more fibers in the direction of maximum stresscaused due to the presence of boundary condition effects.

The inclusion of woven fabric in the filament wound composite pipes maybe beneficial if the woven fabric selected has a higher tensile strengththan the unidirectional glass or carbon fibers.

Optimization of Design Parameters

The optimization of the impact resistance of the composite plates andpipes against low velocity impact loads is important in terms of anumber of advantages. Optimized solutions are lighter in terms of weighthence saving materials and resulting in low cost efficient products.Generally, optimization is performed on a selected function commonlytermed as the cost function which is the function of several variables.The cost function, if properly defined, may be used with a variety oftechniques of optimization. The basic optimization idea is to minimizeor maximize this cost function by choosing the input variables in suchmanner that it forms the best possible solution among a set of possiblesolutions. The history of optimization dates back to the first knownoptimization technique of Steepest Descent pioneered by Gauss. With theadvent of last century the available techniques are more refined and nowfind themselves being employed in a multitude of scientific andtechnological fields. Mathematically, the problem is represented as:Optimize y=ƒ(x ₁ ,x ₂ , . . . , x _(n))  (6.1)

$\begin{matrix}{{{Subject}\mspace{14mu}{to}\mspace{14mu}{g_{j}\left( {x_{1},x_{2},\ldots\mspace{14mu},x_{n}} \right)}\begin{Bmatrix} \leq \\ = \\ \geq \end{Bmatrix}b_{j}}{{j = 1},2,\ldots\mspace{14mu},m}} & (6.2)\end{matrix}$

Optimization Problem

The cost function represented in Eq. (6.1) by ‘y’ is the amount ofabsorbed energy and the cost of the plate or the pipe. The dependentvariables X₁, X₂ etc. are the layer thickness, orientation angles orstacking sequence, number of layers and the material type. There are twoobjectives to minimize simultaneously which makes the problem asmulti-objective optimization, but the objectives here are notcontradictory, therefore, may be combined in one single function.

There are a number of optimization techniques available as described inthe literature review section. Any optimization technique is based uponthe cost function, which in this case is not defined analytically. Toget the cost function, models like linear regression model or othersimilar techniques are used. Because the data is not well structured andhas a lot of variations from point to point, regression models wereunable to predict the empirical mathematical equation. To obtain afunction that may predict the amount of absorbed energy which will thenbe used as the cost function, artificial neural networks were utilized.The ANN model available with the commercially available softwareMATLAB®, the ANN model may be used for function fitting of highlynon-linear data. This technique was then used and optimized to get thebest possible model that may predict the amount of absorbed energy.

Artificial Neural Networks

Artificial Neural Network (ANN) or sometimes called Neural Network is aninterconnected group of artificial neurons that uses a mathematicalmodel or computational model for information processing based on aconnectionist approach to computation. It is an adaptive system whosestructure is modifiable based on the external or internal informationthat flows through the network. The name is given because of its abilityto learn like human brain by examples. This technique is useful inpattern recognition, model fitting or data classification. Once trained,ANN may be used to predict the outcome of new independent data differentfrom the training set. The ability of ANN model to learn by examplehighly non-linear and noisy data is useful in this approach wherestatistical data is dealt with. This feature is very useful in thisproblem where a mathematical relationship of the factors considered bysensitivity analysis with the absorbed impact energy is not availablebut with the help of FEA simulations a lot of training data is availableto us.

A neural network is a set of connected neurons, these neurons receiveimpulses from either input cells or other neurons and apply a functionand transmit the output to other neurons or the final output cells. Theneural networks may be multi-layered in which case one layer receivesinformation from the preceding layer of neurons and passes the output tothe subsequent layers.

A neuron is a real function of the input vector (y₁, y₂, . . . , y_(k)).The output is a function described as:

$\begin{matrix}{{f\left( x_{j} \right)} = {f\left( {\alpha_{j} + {\sum\limits_{i = 1}^{k}\;{w_{ij}y_{j}}}} \right)}} & (6.3)\end{matrix}$

Where, ƒ is a typically a function as sigmoid (log or tan h) function. Agraphical representation of neuron is illustrated in FIG. 29.

Feed Forward Networks

A feed forward network works in the forward direction i.e. the flow ofinformation is in only one direction along the connections from theinput layer through the hidden layers of neurons to the final outputlayer. There is no feedback loop in these networks and hence the outputdoes not affect the performance of the previous layers or the samelayer. FIG. 30 illustrates a multi-layered feed forward artificialneural network.

ANN Model for Flat Plates

Two separate ANN models were generated for the carbon/epoxy and theglass/epoxy plates. In total there were 108 different simulation datafor each type of material.

Data set available for training ANN in this study is 108 samples, fewiterations of ANN models were tried coupled with a differentialevolution algorithm for the optimization of the ANN model in terms ofthe number of neurons and the hidden layers. The data set was randomlydistributed in three sets, for the training, testing and validation ofthe model. The training was carried out by randomly selecting 94 datapoints and the rest were divided equally for the testing and validation.

The optimization algorithm of differential evolution was used to findthe best ANN model, an objective function was defined which computes themaximum error from one ANN model at a time which was based on the numberof neurons. This optimization of the ANN model was necessary to find thebest possible configuration of ANN models which depend upon the numberof hidden layers and neurons. The ANN model configuration thus obtainedwas then train to predict the amount of absorbed energy for thecomposite plates. Two separate models were used to predict the behaviorof composite plates based on carbon or glass fibers.

The carbon/epoxy composite plates' impact behavior was well definedcompared to the glass/epoxy composite plates. It is noted that the morethe data follows a pattern, the better the correlation will be, as theANN model described earlier uses the target response to calculate theweights of each neurons. The model for carbon/epoxy plates needed only21 neurons and a single hidden layer containing all the neurons. Themodel is generally supposed to predict the behavior accurately when theabsolute error between the predicted and the targeted values is at least2 orders less in magnitude.

The ANN model for carbon/epoxy system has a root mean square error ofjust 0.08 J with the maximum error of 0.6242 J.

TABLE 33 Testing ANN for 21 neurons for CFRP plates. Input1 Input2Input3 Actual Simulated (thickness (Number (Stacking response Responsemm) of Layers) Sequence) (Abaqus) J (ANN) J Difference 0.3 20 1 5.91556.105 −0.1895 0.25 16 1 6.7855 6.8955 −0.11 0.25 20 3 5.7807 5.62170.159 0.35 28 1 2.7507 2.5044 0.2463 0.16 16 3 7.5475 7.3142 0.2333 0.316 1 6.5986 6.4669 0.1317 0.25 20 1 7.074 7.0582 0.0158 0.18 16 1 7.71218.0439 −0.3318 0.25 28 1 5.7693 5.7452 0.0241 0.18 16 4 7.1083 7.080.0283 0.4 28 3 1.2529 1.3213 −0.0684 0.4 32 1 0.2346 0.0313 0.2033 0.224 4 4.631 4.7639 −0.1329 0.25 20 2 5.3035 5.3961 −0.0926

A separate verification was carried out with simulations from ABAQUS andthe ANN model for the cases presented in Table 34. The verificationgives the further confidence in the ANN model and its use in generatingthe population for the optimization process. FIG. 31A shows thecorrelation between the target and the predicted response while FIG. 31Brepresents the difference between the actual and the predicted response.

TABLE 34 Independent test cases to verify ANN model. Input1 Input2Input3 Actual Simulated (thickness (No. of (Stacking response Responsemm) Layers) Sequence) (Abaqus) J (ANN) J Difference 0.24 24 1 5.90065.9836 −0.083 0.16 30 4 4.6322 5.0069 −0.3747 0.22 26 2 4.6903 4.9347−0.2444 0.14 18 3 7.7589 8.5684 −0.8095 0.36 32 2 0.5093 0.6132 −0.1039

A similar ANN model was trained to predict the glass/epoxy compositeplates. The ANN model for glass fiber plates uses 24 neurons in a singlelayer and is able to predict the amount of absorbed energy with maximumerror of 1.1047 J and root mean square error of 0.33 J.

TABLE 35 Testing ANN for 24 neurons for GFRP plates. Input1 Input2Input3 Actual Simulated (thickness (Number (Stacking response Responsemm) of Layers) Sequence) (Abaqus) J (ANN) J Difference 0.6 28 1 1.60221.653 1 −0.0509 0.25 32 3 14.8979 14.9839 −0.086 0.4 28 3 11.173211.0837 0.0895 0.25 32 1 13.8011 12.9575 0.8436 0.45 36 4 2.5115 2.49350.018 0.35 36 3 8.6502 8.2851 0.3651 0.4 36 1 8.5923 8.8273 −0.235 0.428 2 14.9456 14.0875 0.8581 0.35 32 4 5.187 5.7195 −0.5325 0.45 32 38.2871 8.5516 −0.2645 0.5 36 2 0.7561 0.7411 0.015 0.3 24 1 12.68612.8399 −0.1539 0.25 36 4 6.3016 5.7934 0.5082 0.35 24 4 11.046 11.6601−0.6141

FIG. 32A illustrates a correlation between a predicted response and atarget response for GFRP plates. FIG. 32B illustrates scatter data of anactual response and vs. a predicted response for GFRP plates.

TABLE 36 Independent test cases to verify ANN model for GFRP plates.Input1 Input2 Input3 Actual Simulated (thickness (No. of (Stackingresponse Response mm) Layers) Sequence) (Abaqus) J (ANN) J Difference0.26 24 2 13.37359 15.5417 2.1681 0.42 30 4 6.14928 6.2809 0.1316 0.3526 1 12.69973 13.153 0.4532 0.54 34 2 0.602984 0.6865 0.0835 0.36 36 38.449098 8.3282 −0.1209

The number of data samples for training is the same for carbon and glassfiber plates but the error is more pronounced for the glass fiber platesdue to the reason that the data for the response is not following apattern which makes it harder to model ANN. This error may be reduced byintroducing more data for training purposes.

ANN Model for Pipes

The training of ANN models for composite pipes is trickier as may beobserved from the graphs presented earlier. It may be observed that inmost of the cases for carbon and glass fiber pipes, there is a rangewhere the absorbed energy value remains more or less the same and thenit decreases suddenly. This sudden change is modeled using more neuronsin the ANN models. In total there are 162 points for the training andvalidation which is almost 1.5 times that of the plates.

For the carbon fiber pipes, the ANN model is particularly worse in thecorrelation. Even with 100 neurons distributed a single hidden layer;the root mean square error is as high as 0.32 J and the maximum error isabout 1.49 J.

TABLE 37 Testing ANN for 100 neurons for CFRP pipes. Input1 Input2Input3 Actual Simulated (thickness (Number (Stacking response Responsemm) of Layers) Sequence) (Abaqus) J (ANN) J Difference 0.425 55 360.3935 1.3454 −0.9519 0.35 55 32 6.646 6.3746 0.2714 0.3 45 16 11.960311.9681 −0.0078 0.4 55 28 7.1349 6.9401 0.1948 0.4 65 32 4.3499 4.08340.2665 0.35 57.5 20 10.6618 9.8097 0.8521 0.25 75 32 11.1477 11.1986−0.0509 0.425 45 36 0.5237 0.5876 −0.0639 0.35 75 32 8.5637 7.654 0.90970.25 57.5 24 11.1164 9.6288 1.4876 0.3 35 28 11.3783 11.3015 0.0768 0.2545 28 10.5071 10.5036 0.0035 0.25 52.5 24 11.3505 11.233 0.1175 0.3 6528 8.6457 8.3328 0.3129

FIG. 33A illustrates a correlation between a predicted response and atarget response for CFRP pipes. FIG. 33B illustrates scatter data of anactual response and vs. a predicted response for CFRP pipes.

The prediction performance of ANN model for the glass fiber pipes ismuch better with a maximum error of 1.026 J and root mean square errorof only 0.26 J. These results from ANN model are not accurate enough butthe absolute error in most of the cases is small enough to consider themodel for prediction. The numbers of neurons used in this model are 37and the number of hidden layer is 1.

TABLE 38 Testing ANN for 37 neurons for GFRP pipes. Input1 Input2 Input3Actual Simulated (thickness (Number of (Stacking response Response mm)Layers) Sequence) (Abaqus) J (ANN) J Difference 0.3 55 40 9.5385 9.6133−0.0748 0.35 65 28 11.1121 11.1812 −0.0691 0.35 65 20 11.9186 11.85750.0611 0.25 65 36 11.0703 11.0599 0.0104 0.375 65 36 6.7361 6.71120.0249 0.35 55 36 8.6462 9.2142 −0.568 0.4 50 28 11.5286 11.5219 0.00670.3 57.5 28 11.4357 11.6259 −0.1902 0.35 75 40 7.9836 8.1484 −0.16480.25 35 32 12.4234 12.1281 0.2953 0.25 57.5 28 12.0485 12.1576 −0.10910.425 35 36 0.8283 0.6212 0.2071 0.35 75 32 10.4983 10.4393 0.059 0.2565 28 11.7907 11.7143 0.0764

FIG. 34A illustrates a correlation between a predicted response and atarget response for GFRP pipes. FIG. 34B illustrates scatter data of anactual response and vs. a predicted response for GFRP pipes.

Cost Models

Composite materials and their production is an expensive process. It hasalways been the focus of major design and development teams to reducethe costs while simultaneously achieve maximum performance. The idea forthis study is optimizing the impact performance with respect to thecosts. To estimate the costs related to the composite plates and pipes,it is necessary to develop a cost model which may relate the costs ofthe material and the production with the samples. A simple yet realisticcost model is proposed in this section, the cost model adopted here isgiven by:CF=X+[(C1/100)+(C2/100)]*X  (6.4)

In this equation, CF represents the total costs, whereas X is assumed tobe the material costs. In general, material costs are considered to bethe maximum and the other costs like labor costs ‘C1’ and the otheroverheads ‘C2’ are considered to be some fraction of the material costs.

An online survey for the prices of the different types of fibers gave abasic idea of the material costs. The prices listed in the Table 39 arefor a reference and may vary depending upon a number of factors rangingfrom the supplier to the texture of the fiber.

TABLE 39 Material costs of different types of fibers. Material TypePrice Carbon fiber Woven fabric 200 USD per m² Glass fiber Woven fabric12 USD per m² Carbon fiber Unidirectional 900 USD per kg Glass fiberUnidirectional 30 USD per kg

Based on these prices for the materials used in the manufacturing ofcomposite plates and pipes, it is obvious that the optimization withrespect to the cost is important.

Differential Evolution Algorithms

Differential evolution algorithms were developed in mid 90s as anoptimization technique by Rainer Storn and Kenneth Price. It is a simpleand robust population based optimization technique with few controlvariables and fast convergence. Being an evolutionary algorithm, the DEtechnique is suited for solving non-linear and non-differentiableoptimization problems. DE is a kind of search technique which works onfinding the candidate solution among a population. DE algorithmsgenerate new populations from the existing one based on certainparameters like mutations and crossovers. The details about thedifferential evolution algorithm are not discussed here. For thisproblem, an initial population size of 200, with a crossover of 0.8 anda total of 100 generations is used to find the optimal solution.

Cost Optimization

The results from the all the analysis as discussed in previous chaptersindicate that the improvement in impact resistance is not linearlydependent on the factors considered. Thus, it is necessary to study thecost optimization of both the composite plates and the composite pipes.A differential evolution algorithm was adopted to optimize the amount ofabsorbed energy by the plate or the pipe and the cost model was used topredict the cost of making that sample.

Separate optimizations for the CFRP plates and the GFRP plates wascarried out and compared with the costs for both types of materials. Itis assumed here that in addition to the cost and the impact performance,the weight of the structure and the thickness of the plate should alsobe a factor in finding the best compromise.

For GFRP plates, a series of runs of the optimization algorithm, it wasfound that the optimal solution is a plate having 36 number of layersusing stacking sequence 4 with the thickness of each layer to be about0.57 mm. At this configuration, the ANN model predicts the absorbedenergy by the plate to be 0.004 J. The weight of the composite platewith this configuration is 0.56 kg and assuming the price listed inTable 39, the cost is estimated to be 14 USD. But it is known that thesheets of 0.57 mm may not be available commercially while 0.6 mm thickwoven fabrics are available. Therefore, a design of composite plate withglass fiber to be used with 0.6 mm thick layers and 36 layers in totalwith the stacking sequence 4 is proposed. This configuration will weighabout 0.59 kg and cost of 14.75 USD.

Similarly, for CFRP plates, the optimal solution was found to be platewith 32 layers of stacking sequence 1 and the thickness of each layer to0.38 mm. This configuration will weigh about 0.29 kg and the amount ofabsorbed energy as predicted by the ANN model is 0.102 J. The cost ofthis plate would be around 260 USD which is a lot as compared to just 14USD for the glass plate although it saves almost half of the weight ofthe glass plate. Simulations were performed for both CFRP plates and theGFRP plates with absorbed energy of 0.21 J and 0.31 J respectively. Theresults show that the ANN prediction model and optimization algorithmperforms well.

As concluded earlier, a best configuration would be to use the stackingsequence 4 with mainly GFRP layers and replacing top and middle layerswith the CFRP layers. This configuration is believed to perform betterin terms of less weight and thickness with some increase in price.

Also, the same optimization procedure is applied to the composite pipes.The results from ANN in this case have some error but the procedure ingeneral is applicable. This ANN model may skip some of the betterresults but due to the error in estimation of the absorbed energy, theoptimization algorithm would reject a better one in favor of a worse butreliable solution. About 10 runs of optimization algorithm wereperformed and the most repeated configuration was selected. It was foundfrom the optimization routine for the carbon fiber pipes; the bestsolution would be to have winding angle of 42.5° with total 36 layersand having each layer of 0.425 mm thick. According to ANN model thisconfiguration would absorb energy of about 0.2 J. The price estimate forthis type of pipe is about 1370 USD while the weight is about 6.8 kg. Asimulation was performed in ABAQUS for this configuration which reportsabsorbed energy of 0.3 J.

Optimization for the glass fiber pipes suggested that optimal solutionto be pipes with winding angle of 51° with total of 40 layers and eachlayer of about 0.4 mm thick. According to the ANN model thisconfiguration would absorb around 0.965 J of energy. The cost estimatefor this configuration of pipe is about 250 USD. This pipe weighs around9 kgs. Simulation of GFRP pipe with the optimized configuration usingABAQUS suggests the results are quite accurate as the predicted absorbedenergy is close to the one from simulation which is 0.88 J.

A simulation in the ABAQUS environment of the proposed solution from theoptimization algorithm confirmed the results for all the cases. As wasthe case with plates, the compromise between price and the weight may beachieved by replacing top layers of GFRP pipes with the carbon fiberlayers as previously discussed.

The main conclusions from this section may be summarized as:

ANN models are very strong and useful tools for the function fitting ofnon-linear behavior and as observed in the case of CFRP plates are ableto predict the absorbed energy with very little error.

The accuracy of the ANN models depend upon the behavior of the trainingdata sets, if there are too much sudden variations in the training dataas was observed in the results from the composite pipes then the modelmay be prone to errors.

A better way to model ANN with training data as in this case is tosimulate and generate a very big training data. In this study, therewere around 100 data each for the flat plates and about 150 for thepipes apart from the ones that were used to validate the results. As arule of thumb, it is suggested that the data size should be in the rangeof 500-1000 for a very accurate model.

The results and discussions about the findings were already discussed indetail with each section. The following may be summarized as theconclusions:

The Hashin damage model used as the damage initiation model in thisresearch is accurate enough to predict the onset of damage without lossof much accuracy.

Sensitivity analysis is a useful tool in determining the factorsinfluencing the most on the impact performance of the FRP plates andpipes.

The amount of absorbed energy considered as an indication for the amountof damage is affected mainly by the thickness of the layers, number oflayers, stacking sequence and the material properties.

Material properties like the tensile strength of the fiber and thefracture energies of the laminate during the tensile loading in thelongitudinal direction are more influential than other mechanicalproperties of the fiber or the binder material used.

Quasi isotropic laminates show good performance in all conditions. But,the stacking sequence other than quasi isotropic laminates may haveoptimal performance if the boundary conditions are such that theyrestrict deformation in any particular direction and allow in the otherdirections.

Influence of boundary conditions may be controlled by aligning morelayers to counter the stress produced as a result of bending.

The numbers of layers also have an effect on the impact resistance andshould be selected carefully as to not just increase the layers whichresult in more contact force and hence greater damage.

Design of experiment is a useful method to statistically study thevariation in the impact performance of the FRP plates and pipes withrespect to the variables identified using sensitivity analysis.

The effect of thickness of the laminate is the most interesting onecompared to the other factors. The thickness is not directlyproportional to the impact performance of the plates or the pipes. Thinplates have better performance compared to plates that are thick but notrigid enough.

The amount of dissipated energy transferred to the specimen is notalways in the form of damage dissipation but some of the energy istransferred to the specimen which generates unnecessary vibration. Basedon the above two conclusions, the specimen thickness should be such thatit is stiff and thick enough to withstand the impact loads withoutsuffering from the vibrations.

It is important to have as more fibers as possible in the direction ofthe maximum stress during the impact to delay the damage initiationprocess.

The design of the structure and the choice of material should be suchthat the maximum utilization of the tensile strength of the fibermaterials may be achieved.

The inclusion of woven fabric in the filament wound composite pipes maybe beneficial if the woven fabric selected has a higher tensile strengththan the unidirectional glass or carbon fibers.

Using the optimization algorithm, it was suggested that the optimalstacking sequence for the flat plates would be the sequence number 4from this study.

The inclusion of carbon fibers in the flat plates and the pipes mayenhance the impact resistance quite a lot with the added advantage ofweight savings as well as reduced thickness at a slightly higher price.

Obviously, numerous modifications and variations of the presentinvention are possible in light of the above teachings. It is thereforeto be understood that within the scope of the appended claims, theinvention may be practiced otherwise than as specifically describedherein.

A method in accordance with an exemplary implementation of presentinvention is illustrated in FIG. 35. As illustrated in FIG. 35, a methodfor predicting an impact resistance of a composite material may begin inStep 10.

In Step 10, an artificial neural network may be designed. Discussionrelating to artificial neural networks has been previously presentedrelating to FIG. 30. In an exemplary implementation of the presentinvention, the artificial neural network may be designed to include aplurality of layers, each layer including a plurality of neurons. Theartificial neural network may be a feed forward network. As illustratedin FIG. 30, an artificial neural network may be designed to include aninput layer and an output layer. Additionally, the artificial neuralnetwork may include one or more hidden layers.

The input layer of the artificial neural network may be designed toinclude a plurality of neurons. Each neuron in the input layer mayreceive data that is input into the artificial neural network. In anexemplary implementation of the present invention, each neuron in theinput layer may perform a process upon data that is input. Afterperformance of the process, each neuron may then output data to one ormore neurons in a next layer of the artificial neural network. In anexemplary implementation, each neuron of the input layer may output datato one or more neurons in the hidden layer.

In an exemplary implementation of the present invention, the artificialneural network further includes a single hidden layer, as illustrated inFIG. 30. However, other exemplary implementations of the presentinvention may include zero or more than one hidden layer within theartificial neural network.

The hidden layer may be designed to include a plurality of neurons. Eachneuron in the hidden layer may receive data from one or more neurons inthe input layer of the artificial neural network. In an exemplaryimplementation of the present invention, each neuron in the hidden layermay perform a process upon the input data. After performance of theprocess, each neuron of the hidden layer may then output data to one ormore neurons in the output layer of the artificial neural network.

The output layer of the artificial neural network may be designed toinclude a plurality of neurons. Each neuron in the output layer of theartificial neural network may receive data from one or more neurons in aprevious layer of the artificial neural network. In an exemplaryimplementation, each neuron in the output layer may receive data fromone or more neurons in the hidden layer. In alternative implementations,each neuron in the output layer may receive data from one or moreneurons in an additional layer within the artificial neural network.

The artificial neural network may be designed according to the type ofdata to be input into the artificial neural network. Data that may beinput into the artificial neural network regarding the compositematerial may include, for example:

stacking sequence of layers in the composite material;

layer thickness;

number of layers in the composite material;

orientation angle of the layers in the composite material; and/or

a material composition of the layers in the composite material.

However, other data relating to the composite material may be input intothe artificial neural network for computation that is required so as topredict an impact resistance of the composite material, as previouslydiscussed. For example, the artificial neural network may receive datarelating to dimensional information of each layer in the compositematerial, dimensional information of the composite material as a whole,and/or the shape and structure of the composite material. Alternatively,the artificial neural network may also be designed in accordance withfactors such as processing power, time, the amount of data required foran accurate prediction, and/or cost.

Additionally, data may be input into the artificial neural network so asto train the artificial neural network to more accurately predict animpact resistance of the composite material. Such data input isdiscussed in Step 20.

In Step 20, the artificial neural network is trained to more accuratelypredict an impact resistance of the composite material. Step 20 mayinclude Steps 22, 24, 26 and/or 28.

In Step 22, sample data is input into the input layer of the artificialneural network. The sample data may include input data relating to asample composite material that has a known impact resistance. That is,the sample data may relate to a sample composite material in which theimpact resistance is already known, and thus, need not be predicted. Thesample data may be used so as to train or tune the artificial neuralnetwork to more accurately predict an impact resistance of the compositematerial.

After the sample data is fed into the artificial neural network in Step22, the artificial neural network processes the sample data. In Step 24,a predicted impact resistance may be output from the artificial neuralnetwork relating to the sample composite material. In Step 26, an errorof the predicted impact resistance may be calculated by measuring adifference between the known impact resistance of the sample compositematerial and the predicted impact resistance that was output from theartificial neural network. In exemplary implementations of the presentinvention, the error may be a mean-squared error. Alternatively, theerror may be calculated by other methods.

After the error is acquired in Step 26, the artificial neural networkmay be trained by reducing an error in the predicted impact resistancecalculated by the artificial neural network. In Step 28, the error maybe reduced by adjusting inputs and outputs of neurons in the artificialneuron network. In particular, the error may be reduced by applying avariable weighting factor to each neuron in the artificial neuralnetwork to adjust an output of each neuron. In other words, an output aneuron may be increased or decreased by a factor so as to increase ordecrease the impact of the neuron's output within the next layer'sprocessing. The variable weighting factor may be any whole or fraction,positive or negative.

Additionally, the error may be reduced by managing what data is inputinto each neuron in the hidden layer. For example, data output fromneurons in the input layer may be selected for input to a neuron in thehidden layer. In other words, for each neuron in the hidden layer, dataoutput from neurons in the input layer may be input or may not be input.Further, data input into each neuron within the hidden layer may beindividually managed such that individual neurons in the hidden layermay receive input from different input layer neurons. As a result, datathat is input into each neuron of the hidden layer may be managed so asto manage a processing of each neuron in the hidden layer.

In implementations of the present invention that include more than onehidden layer, or include other additional layers, inputs of data forneurons in each additional hidden layer or other layer may beindividually managed so as to manage a processing of each neuron in eachlayer.

Furthermore, the error may be reduced by both (1) adjusting inputs andoutputs of neurons in the artificial neuron network, and (2) managingwhat data is input into each neuron in the hidden layer. However, inother exemplary implementations, the error may be reduced byalternatively (1) adjusting inputs and outputs of neurons in theartificial neuron network, or (2) managing what data is input into eachneuron in the hidden layer.

Once the inputs and outputs of neurons in the artificial neural networkhave been modified, Steps 22, 24, 26 and 28 may be repeated multipletimes with the same sample data and/or different sample data so as tofurther train the artificial neural network.

After the artificial neural network has been trained to predict animpact resistance of a composite material, the method proceeds to Step30. In Step 30, data relating to the composite material may be inputinto the artificial neural network. The data may be input in a mannersimilar or analogous to the input of data in Step 22.

After data relating to the composite material has been input into theartificial neural network, the artificial neural network processes theinput data. The method then proceeds to Step 40. In Step 40, a predictedimpact resistance may be calculated and output from the artificialneural network relating to the composite material.

The method as previously described and illustrated in FIG. 35 may befurther utilized with the artificial neural network to predict anoptimized design of a plate or pipe comprising the composite material.

An alternative implementation of the present invention may include acomputer readable medium that stores computer readable instructionsthat, when executed by a computer, may cause the computer to perform themethod for predicting an impact resistance of a composite material, aspreviously described and illustrated in FIG. 35.

In another exemplary implementation of the present invention, a devicecomprising a processor may be configured to perform the method forpredicting an impact resistance of a composite material, as describedabove and illustrated in FIG. 35. Such a device is illustrated in FIG.36.

In FIG. 36, the device includes a CPU 100 which may perform the methoddescribed above and illustrated in FIG. 35. The process data andinstructions may be stored in memory 102. These processes andinstructions may also be stored on a storage medium disk 104 such as ahard drive (HDD) or portable storage medium or may be stored remotely.Further, the claimed advancements are not limited by the form of thecomputer-readable media on which the instructions of the inventiveprocess are stored. For example, the instructions may be stored on CDs,DVDs, in FLASH memory, RAM, ROM, PROM, EPROM, EEPROM, hard disk or anyother information processing device with which the device communicates,such as a server or computer.

Further, the claimed advancements may be provided as a utilityapplication, background daemon, or component of an operating system, orcombination thereof, executing in conjunction with CPU 100 and anoperating system such as Microsoft Windows 7, UNIX, Solaris, LINUX,Apple MAC-OS and other systems known to those skilled in the art.

CPU 100 may be a Xenon or Core processor from Intel of America or anOpteron processor from AMD of America, or may be other processor typesthat would be recognized by one of ordinary skill in the art.Alternatively, the CPU 100 may be implemented on an FPGA, ASIC, PLD orusing discrete logic circuits, as one of ordinary skill in the art wouldrecognize. Further, CPU 100 may be implemented as multiple processorscooperatively working in parallel to perform the instructions of theinventive processes described above.

The device in FIG. 36 also includes a network controller 106, such as anIntel Ethernet PRO network interface card from Intel Corporation ofAmerica, for interfacing with a network. As can be appreciated, thenetwork can be a public network, such as the Internet, or a privatenetwork such as an LAN or WAN network, or any combination thereof andcan also include PSTN or ISDN sub-networks. The network can also bewired, such as an Ethernet network, or can be wireless such as acellular network including EDGE, 3G and 4G wireless cellular systems.The wireless network can also be WiFi, Bluetooth, or any other wirelessform of communication that is known.

The device further includes a display controller 108, such as a NVIDIAGeForce GTX or Quadro graphics adaptor from NVIDIA Corporation ofAmerica for interfacing with display 110, such as a Hewlett PackardHPL2445w LCD monitor. A general purpose I/O interface 112 interfaceswith a keyboard and/or mouse 114 as well as a touch screen panel 116 onor separate from display 110. General purpose I/O interface alsoconnects to a variety of peripherals 118 including printers andscanners, such as an OfficeJet or DeskJet from Hewlett Packard.

The device may be utilized to output the predicted impact resistance.Once the predicted impact resistance is output from the artificialneural network, components of the device may output the prediction to arecipient. For example, display 110, touch panel 116, network controller106 and/or speakers/microphone 122 may output the prediction to arecipient. Network controller 106 may output the prediction over thenetwork. Further, the prediction may be stored locally within memory102, storage medium disk 104, or recorded in another removable internalor external storage medium.

A sound controller 120 is also provided in the device, such as SoundBlaster X-Fi Titanium from Creative, to interface withspeakers/microphone 122 thereby providing sounds and/or music.

The general purpose storage controller 124 connects the storage mediumdisk 104 with communication bus 126, which may be an ISA, EISA, VESA,PCI, or similar, for interconnecting all of the components of thedevice. A description of the general features and functionality of thedisplay 110, keyboard and/or mouse 114, as well as the displaycontroller 108, storage controller 124, network controller 106, soundcontroller 120, and general purpose I/O interface 112 is omitted hereinfor brevity as these features are known.

In various implementations of the present invention, the artificialneural network may be located over a network and accessible by thedevice via network controller 106. Alternatively, the artificial neuralnetwork may be accessed by the device via I/O interface 112. In otherexemplary implementations of the present invention, the functionality ofthe artificial neural network may be executed by CPU 100.

Additionally, an exemplary implementation of the present invention mayinclude a system for predicting an impact resistance of a compositematerial. In such a configuration, the system may include a device, aspreviously discussed and illustrated in FIG. 36, the device executingthe method as previously discussed and illustrated in FIG. 35, as wellas the artificial neural network.

The foregoing discussion discloses and describes merely exemplaryimplementations of the present invention. As will be understood by thoseskilled in the art, the present invention may be embodied in otherspecific forms without departing from the spirit or essentialcharacteristics thereof. Accordingly, the disclosure of the presentinvention is intended to be illustrative, but not limiting of the scopeof the invention, as well as other claims. The disclosure, including anyreadily discernible variants of the teachings herein, define, in part,the scope of the foregoing claim terminology such that no inventivesubject matter is dedicated to the public.

APPENDICES A.1 Results for the Composite Plates

TABLE A. 1 List of experiments numerically solved for the layerconfiguration 1 for Carbon/Epoxy plates Layer Total Rebound ThicknessNumber Thickness of Velocity Absorbed S. No. (mm) of Layers Plate (mm)(m/s) Energy (J) 1 0.12 16 1.92 3.7716 16.8809 2 0.14 16 2.24 4.540312.0893 3 0.16 16 2.56 4.9512 9.1643 4 0.18 16 2.88 5.1430 7.71212 50.12 20 2.4 4.5898 11.7503 6 0.16 20 3.2 5.0757 8.2276 7 0.18 20 3.65.0685 8.2824 8 0.2 16 3.2 5.1193 7.8947 9 0.25 16 4 5.2618 6.7855 100.3 16 4.8 5.2854 6.5986 11 0.2 20 4 5.1614 7.5701 12 0.25 20 5 5.22517.0740 13 0.3 20 6 5.3709 5.9155 14 0.2 24 4.8 5.2905 6.5578 15 0.25 246 5.3791 5.8493 16 0.3 24 7.2 5.3403 6.1606 17 0.2 28 5.6 5.3385 6.175618 0.25 28 7 5.3890 5.7693 19 0.3 28 8.4 5.5732 4.2549 20 0.35 28 9.85.7503 2.7507 21 0.4 28 11.2 5.9987 0.56125 22 0.3 32 9.6 5.7284 2.939123 0.35 32 11.2 6.0384 0.2031 24 0.4 32 12.8 6.0349 0.2346 25 0.3 3610.8 5.9588 0.9193 26 0.35 36 12.6 6.0409 0.18073 27 0.4 36 14.4 6.04860.11064

TABLE A. 2 List of experiments numerically solved for the layerconfiguration 2 for Carbon/Epoxy plates Layer Total Rebound ThicknessNumber Thickness of Velocity Absorbed S. No. (mm) of Layers Plate (mm)(m/s) Energy (J) 1 0.12 16 1.92 4.11064 14.8770 2 0.14 16 2.24 4.6988910.9903 3 0.16 16 2.56 5.06889 8.2798 4 0.18 16 2.88 5.14636 7.6862 50.12 20 2.4 4.96802 9.0391 6 0.16 20 3.2 5.2784 6.6539 7 0.18 20 3.65.37821 5.8561 8 0.2 16 3.2 5.26623 6.7501 9 0.25 16 4 5.40544 5.6359 100.3 16 4.8 5.42364 5.4881 11 0.2 20 4 5.41088 5.5918 12 0.25 20 55.44628 5.3035 13 0.3 20 6 5.49541 4.9003 14 0.2 24 4.8 5.39271 5.739015 0.25 24 6 5.50199 4.8461 16 0.3 24 7.2 5.46472 5.1526 17 0.2 28 5.65.51426 4.7447 18 0.25 28 7 5.43048 5.4324 19 0.3 28 8.4 5.59542 4.068520 0.35 28 9.8 5.79284 2.3823 21 0.4 28 11.2 5.97076 0.8125 22 0.3 329.6 5.82705 2.0841 23 0.35 32 11.2 5.97504 0.7742 24 0.4 32 12.8 6.036070.2244 25 0.3 36 10.8 5.93021 1.1745 26 0.35 36 12.6 6.03809 0.2061 270.4 36 14.4 5.92846 1.1900

TABLE A. 3 List of experiments numerically solved for the layerconfiguration 3 for Carbon/Epoxy plates Layer Total Rebound ThicknessNumber Thickness of Velocity Absorbed S. No. (mm) of Layers Plate (mm)(m/s) Energy (J) 1 0.12 16 1.92 4.02354 15.4083 2 0.14 16 2.24 5.001518.7887 3 0.16 16 2.56 5.1643 7.5475 4 0.18 16 2.88 5.23933 6.9621 5 0.1220 2.4 5.00133 8.7900 6 0.16 20 3.2 5.30247 6.4629 7 0.18 20 3.6 5.293926.5308 8 0.2 16 3.2 5.21854 7.1251 9 0.25 16 4 5.21853 7.1252 10 0.3 164.8 5.44782 5.2909 11 0.2 20 4 5.37119 5.9127 12 0.25 20 5 5.387555.7807 13 0.3 20 6 5.50416 4.8282 14 0.2 24 4.8 5.44647 5.3020 15 0.2524 6 5.50308 4.8371 16 0.3 24 7.2 5.37818 5.8564 17 0.2 28 5.6 5.466565.1375 18 0.25 28 7 5.42745 5.4571 19 0.3 28 8.4 5.59528 4.0696 20 0.3528 9.8 5.64996 3.6085 21 0.4 28 11.2 5.92138 1.2529 22 0.3 32 9.65.62214 3.8437 23 0.35 32 11.2 5.91947 1.2699 24 0.4 32 12.8 6.020440.3657 25 0.3 36 10.8 5.87043 1.7035 26 0.35 36 12.6 6.02154 0.3560 270.4 36 14.4 5.93517 1.1303

TABLE A. 4 List of experiments numerically solved for the layerconfiguration 4 for Carbon/Epoxy plates Layer Total Rebound ThicknessNumber Thickness of Velocity Absorbed S. No. (mm) of Layers Plate (mm)(m/s) Energy (J) 1 0.12 16 1.92 3.42872 18.7329 2 0.14 16 2.24 3.9828815.6525 3 0.16 16 2.56 5.09185 8.1048 4 0.18 16 2.88 5.22069 7.1083 50.12 20 2.4 4.60535 11.6431 6 0.16 20 3.2 5.21299 7.1686 7 0.18 20 3.65.31472 6.3653 8 0.2 16 3.2 5.2942 6.5286 9 0.25 16 4 5.46716 5.1326 100.3 16 4.8 5.522 4.6806 11 0.2 20 4 5.37021 5.9206 12 0.25 20 5 5.551844.4328 13 0.3 20 6 5.61302 3.9205 14 0.2 24 4.8 5.52799 4.6310 15 0.2524 6 5.63551 3.7308 16 0.3 24 7.2 5.56273 4.3420 17 0.2 28 5.6 5.627263.8005 18 0.25 28 7 5.543 4.5064 19 0.3 28 8.4 5.75417 2.7171 20 0.35 289.8 5.77615 2.5271 21 0.4 28 11.2 5.91357 1.3223 22 0.3 32 9.6 5.783992.4591 23 0.35 32 11.2 5.9137 1.3211 24 0.4 32 12.8 6.02662 0.3099 250.3 36 10.8 5.86668 1.7365 26 0.35 36 12.6 6.02121 0.3588 27 0.4 36 14.45.94022 1.0853

TABLE A. 5 List of experiments numerically solved for the layerconfiguration 1 for Glass/Epoxy plates Layer Total Rebound ThicknessNumber Thickness of Velocity Absorbed S. No. (mm) of Layers Plate (mm)(m/s) Energy (J) 1 0.25 24 6 4.29654 13.7048 2 0.3 24 7.2 4.4518212.6860 3 0.35 24 8.4 4.38999 13.0960 4 0.4 24 9.6 4.25182 13.9915 50.45 24 10.8 4.1855 14.4112 6 0.5 24 12 4.01153 15.4807 7 0.6 24 14.45.00643 8.7517 8 0.25 28 7 4.59771 11.6958 9 0.3 28 8.4 4.46418 12.603310 0.35 28 9.8 4.32582 13.5155 11 0.4 28 11.2 4.14263 14.6790 12 0.25 328 4.28157 13.8011 13 0.3 32 9.6 4.26511 13.9066 14 0.35 32 11.2 4.1343314.7305 15 0.25 36 9 4.46612 12.5903 16 0.3 36 10.8 4.09504 14.9730 170.35 36 12.6 3.80239 16.7064 18 0.4 32 12.8 4.51516 12.2600 19 0.4 3614.4 5.02762 8.5923 20 0.45 28 12.6 3.97484 15.7005 21 0.45 32 14.44.99326 8.8505 22 0.45 36 16.2 5.81877 2.1564 23 0.5 28 14 4.986138.9039 24 0.5 32 16 5.80594 2.2683 25 0.5 36 18 5.97269 0.7952 26 0.6 2816.8 5.88193 1.6022 27 0.6 32 19.2 6.01784 0.3892

TABLE A. 6 List of experiments numerically solved for the layerconfiguration 2 for Glass/Epoxy plates Layer Total Rebound ThicknessNumber Thickness of Velocity Absorbed S. No. (mm) of Layers Plate (mm)(m/s) Energy (J) 1 0.25 24 6 4.3052 13.6489 2 0.3 24 7.2 4.18788 14.39623 0.35 24 8.4 4.46527 12.5960 4 0.4 24 9.6 4.35886 13.3003 5 0.45 2410.8 4.15121 14.6256 6 0.5 24 12 3.86215 16.3628 7 0.6 24 14.4 5.673963.4046 8 0.25 28 7 4.3385 13.4331 9 0.3 28 8.4 4.61759 11.5584 10 0.3528 9.8 4.35215 13.3441 11 0.4 28 11.2 4.0995 14.9456 12 0.25 32 84.47352 12.5407 13 0.3 32 9.6 4.38341 13.1393 14 0.35 32 11.2 4.1003514.9403 15 0.25 36 9 4.47352 12.5407 16 0.3 36 10.8 4.17267 14.4916 170.35 36 12.6 4.44619 12.7235 18 0.4 32 12.8 4.62489 11.5078 19 0.4 3614.4 5.67654 3.3827 20 0.45 28 12.6 4.4747 12.5328 21 0.45 32 14.45.6774 3.3753 22 0.45 36 16.2 5.83351 2.0276 23 0.5 28 14 5.70682 3.124224 0.5 32 16 5.81129 2.2217 25 0.5 36 18 5.97706 0.7561 26 0.6 28 16.85.88858 1.5435 27 0.6 32 19.2 6.02185 0.3530

TABLE A. 7 List of experiments numerically solved for the layerconfiguration 3 for Glass/Epoxy plates Layer Total Rebound ThicknessNumber Thickness of Velocity Absorbed S. No. (mm) of Layers Plate (mm)(m/s) Energy (J) 1 0.25 24 6 4.29723 13.7004 2 0.3 24 7.2 4.3062913.6419 3 0.35 24 8.4 4.21859 14.2026 4 0.4 24 9.6 4.02843 15.3788 50.45 24 10.8 4.77313 10.4629 6 0.5 24 12 5.0023 8.7827 7 0.6 24 14.45.05658 8.3732 8 0.25 28 7 4.46288 12.6120 9 0.3 28 8.4 4.40108 13.022910 0.35 28 9.8 4.13868 14.7035 11 0.4 28 11.2 4.67287 11.1732 12 0.25 328 4.10725 14.8979 13 0.3 32 9.6 4.0367 15.3288 14 0.35 32 11.2 4.7340610.7415 15 0.25 36 9 4.30269 13.6651 16 0.3 36 10.8 4.73654 10.7240 170.35 36 12.6 5.01993 8.6502 18 0.4 32 12.8 5.0657 8.3040 19 0.4 36 14.45.07679 8.2197 20 0.45 28 12.6 5.02813 8.5884 21 0.45 32 14.4 5.067938.2871 22 0.45 36 16.2 5.78836 2.4212 23 0.5 28 14 5.08497 8.1573 24 0.532 16 5.76298 2.6410 25 0.5 36 18 5.96128 0.8974 26 0.6 28 16.8 5.861311.7838 27 0.6 32 19.2 6.01275 0.4351

TABLE A. 8 List of experiments numerically solved for the layerconfiguration 4 for Glass/Epoxy plates Layer Total Rebound ThicknessNumber Thickness of Velocity Absorbed S. No. (mm) of Layers Plate (mm)(m/s) Energy (J) 1 0.25 24 6 4.37745 13.1785 2 0.3 24 7.2 3.9540615.8241 3 0.35 24 8.4 4.69099 11.0460 4 0.4 24 9.6 5.37575 5.8760 5 0.4524 10.8 5.43519 5.3940 6 0.5 24 12 5.42735 5.4579 7 0.6 24 14.4 5.681843.3375 8 0.25 28 7 4.1224 14.8044 9 0.3 28 8.4 4.51686 12.2485 10 0.3528 9.8 5.39647 5.7086 11 0.4 28 11.2 5.433 5.4119 12 0.25 32 8 4.5790211.8244 13 0.3 32 9.6 5.37565 5.8768 14 0.35 32 11.2 5.46052 5.1870 150.25 36 9 5.3227 6.3016 16 0.3 36 10.8 5.4226 5.4966 17 0.35 36 12.65.35458 6.0464 18 0.4 32 12.8 5.3718 5.9078 19 0.4 36 14.4 5.675693.3899 20 0.45 28 12.6 5.35868 6.0134 21 0.45 32 14.4 5.67957 3.3569 220.45 36 16.2 5.77795 2.5115 23 0.5 28 14 5.75044 2.7493 24 0.5 32 165.75055 2.7484 25 0.5 36 18 5.95438 0.9590 26 0.6 28 16.8 5.84955 1.887127 0.6 32 19.2 6.01049 0.4555

TABLE A. 9 Combinations for Top and Bottom layers of Carbon/epoxy andResults Carbon Glass Total Num- layer layer Plate ber thick- thick-Thick- of Rebound Absorbed ness ness ness Lay- Velocity Energy No. (mm)(mm) (mm) ers (m/s) (J) 1 0.2 0.35 16 5.3 4.6045 11.6491 2 0.2 0.3 205.8 5.0184 8.6616 3 0.2 0.3 24 7 4.8861 9.6449 4 0.2 0.35 24 8.1 5.32026.3213 5 0.25 0.35 28 9.6 5.3720 5.9063 6 0.25 0.25 32 8 5.3137 6.3734 70.25 0.35 32 11 5.4819 5.0114 1 0.2 0.35 16 5.3 5.0352 8.5348 2 0.2 0.320 5.8 4.4225 12.8813 3 0.2 0.3 24 7 4.7797 10.4161 4 0.2 0.35 24 8.14.8720 9.7481 5 0.25 0.35 28 9.6 5.37442 5.8867 6 0.25 0.25 32 8 4.7725410.4671 7 0.25 0.35 32 11 5.45069 5.2675

TABLE A. 11 Combinations for Top 2 layers of Carbon/epoxy and ResultsCarbon Glass Total Num- layer layer Plate ber thick- thick- Thick- ofRebound Absorbed ness ness ness Lay- Velocity Energy No. (mm) (mm) (mm)ers (m/s) (J) 1 0.2 0.35 16 5.3 4.61488 11.5772 2 0.2 0.25 20 4.94.78974 10.3438 3 0.2 0.3 20 5.8 4.62342 11.5180 4 0.2 0.3 24 7 4.7314410.7601 5 0.2 0.35 24 8.1 5.30862 6.4139 6 0.25 0.35 28 9.6 5.414145.5653 7 0.25 0.25 32 8 5.3265 6.2713 8 0.25 0.35 32 11 5.49488 4.9047

TABLE A. 12 Combinations for Bottom 2 layers of Carbon/epoxy and ResultsCarbon Glass Total Num- layer layer Plate ber thick- thick- Thick- ofRebound Absorbed ness ness ness Lay- Velocity Energy No. (mm) (mm) (mm)ers (m/s) (J) 1 0.2 0.35 16 5.3 3.55864 18.05206 2 0.2 0.25 20 4.93.15915 20.06483 3 0.2 0.3 20 5.8 4.45159 12.68751 4 0.2 0.3 24 74.46224 12.61631 5 0.2 0.35 24 8.1 5.29749 6.50245 6 0.25 0.35 28 9.65.39965 5.682835 7 0.25 0.25 32 8 4.7856 10.37352 8 0.25 0.35 32 115.4845 4.9899

TABLE A. 13 Combinations for Top, Bottom and Middle 2 layers ofCarbon/epoxy Carbon Glass Total Num- layer layer Plate ber thick- thick-Thick- of Rebound Absorbed ness ness ness Lay- Velocity Energy No. (mm)(mm) (mm) ers (m/s) (J) 1 0.2 0.25 16 3.8 5.03532 8.5342 2 0.2 0.35 16 55.1643 7.5479 3 0.2 0.3 20 5.6 5.00095 8.7929 4 0.2 0.3 24 6.8 4.99598.8309 5 0.2 0.35 24 7.8 5.31015 6.4017

TABLE A. 14 Combinations for Top and Middle layers of Carbon/epoxy andResults Carbon Glass Total Num- layer layer Plate ber thick- thick-Thick- of Rebound Absorbed ness ness ness Lay- Velocity Energy No. (mm)(mm) (mm) ers (m/s) (J) 1 0.2 0.35 16 5.3 5.02165 8.6373 2 0.2 0.25 204.9 4.82216 10.1101 3 0.2 0.3 20 5.8 5.08829 8.1320 4 0.2 0.3 24 74.87564 9.7211 5 0.2 0.35 24 8.1 4.9283 9.3339 6 0.25 0.35 28 9.65.39191 5.7455 7 0.25 0.25 32 8 4.96792 9.0398 8 0.25 0.35 32 11 5.49554.8993

A.2 Results for the Composite Pipes

TABLE A. 15 Results of numerical simulation for the Carbon/epoxy pipeshaving 20 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 5 1.88815 22.1744 2 0.3 35 6 2.42007 10.7163 3 0.35 35 7 2.4169910.7908 4 0.4 35 8 2.40259 11.1378 5 0.25 45 5 1.97629 20.4714 6 0.3 456 2.37435 11.8123 7 0.35 45 7 2.42622 10.5673 8 0.4 45 8 2.45385 9.89319 0.25 55 5 2.46942 9.5098 10 0.3 55 6 2.37988 11.6809 11 0.35 55 72.46786 9.5483 12 0.4 55 8 2.43165 10.4354 13 0.25 65 5 2.39083 11.419714 0.3 65 6 2.38887 11.4665 15 0.35 65 7 2.4453 10.1025 16 0.4 65 82.48513 9.1206 17 0.25 75 5 2.422 10.6696 18 0.3 75 6 2.40684 11.0356 190.35 75 7 2.39423 11.3383 20 0.4 75 8 2.40487 11.0830

TABLE A. 16 Results of numerical simulation for the Carbon/epoxy pipeshaving 24 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 6 1.90173 21.9171 2 0.3 35 7.2 2.41621 10.8096 3 0.35 35 8.42.38414 11.5794 4 0.4 35 9.6 2.39278 11.3730 5 0.25 45 6 2.40896 10.98466 0.3 45 7.2 2.44188 10.1861 7 0.35 45 8.4 2.46463 9.6280 8 0.4 45 9.62.42102 10.6933 9 0.25 55 6 2.46441 9.6334 10 0.3 55 7.2 2.47285 9.425111 0.35 55 8.4 2.45131 9.9554 12 0.4 55 9.6 2.47772 9.3045 13 0.25 65 62.38813 11.4842 14 0.3 65 7.2 2.44234 10.1749 15 0.35 65 8.4 2.493578.9105 16 0.4 65 9.6 2.48778 9.0548 17 0.25 75 6 2.40684 11.0356 18 0.375 7.2 2.40243 11.1417 19 0.35 75 8.4 2.37773 11.7320 20 0.4 75 9.62.47919 9.2681

TABLE A. 17 Results of numerical simulation for the Carbon/epoxy pipeshaving 28 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 7 2.41827 10.7599 2 0.3 35 8.4 2.39256 11.3783 3 0.35 35 9.82.38517 11.5548 4 0.4 35 11.2 2.42129 10.6868 5 0.25 45 7 2.4287 10.50716 0.3 45 8.4 2.47588 9.3501 7 0.35 45 9.8 2.46703 9.5688 8 0.4 45 11.22.46637 9.5851 9 0.25 55 7 2.41547 10.8275 10 0.3 55 8.4 2.51334 8.415611 0.35 55 9.8 2.47131 9.4631 12 0.4 55 11.2 2.56379 7.1349 13 0.25 65 72.42896 10.5008 14 0.3 65 8.4 2.50417 8.6457 15 0.35 65 9.8 2.514168.3950 16 0.4 65 11.2 2.61311 5.8583 17 0.25 75 7 2.39331 11.3603 18 0.375 8.4 2.38041 11.6682 19 0.35 75 9.8 2.50703 8.5740 20 0.4 75 11.22.51475 8.3802

TABLE A. 18 Results of numerical simulation for the Carbon/epoxy pipeshaving 32 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 8 2.39549 11.3081 2 0.3 35 9.6 2.39064 11.4242 3 0.35 35 11.22.42538 10.5877 4 0.4 35 12.8 2.54531 7.6070 5 0.25 45 8 2.42531 10.58946 0.3 45 9.6 2.46261 9.6778 7 0.35 45 11.2 2.46798 9.5453 8 0.4 45 12.82.82109 0.2073 9 0.25 55 8 2.49993 8.7517 10 0.3 55 9.6 2.45368 9.897311 0.35 55 11.2 2.58279 6.6460 12 0.4 55 12.8 2.79102 1.0510 13 0.25 658 2.49593 8.8517 14 0.3 65 9.6 2.50256 8.6860 15 0.35 65 11.2 2.618365.7210 16 0.4 65 12.8 2.67021 4.3510 17 0.25 75 8 2.40218 11.1477 18 0.375 9.6 2.48446 9.1373 19 0.35 75 11.2 2.50744 8.5637 20 0.4 75 12.82.58654 6.5491

TABLE A. 19 Results of numerical simulation for the Carbon/epoxy pipeshaving 36 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 9 2.38755 11.4980 2 0.3 35 10.8 2.3883 11.4801 3 0.35 35 12.62.4877 9.0567 4 0.4 35 14.4 2.81955 0.2507 5 0.25 45 9 2.46453 9.6305 60.3 45 10.8 2.42175 10.6756 7 0.35 45 12.6 2.82005 0.2366 8 0.4 45 14.42.81233 0.4540 9 0.25 55 9 2.49248 8.9377 10 0.3 55 10.8 2.5481 7.535911 0.35 55 12.6 2.71305 3.1968 12 0.4 55 14.4 2.77641 1.4577 13 0.25 659 2.49985 8.7537 14 0.3 65 10.8 2.58909 6.4831 15 0.35 65 12.6 2.671484.3160 16 0.4 65 14.4 2.73159 2.6921 17 0.25 75 9 2.45539 9.8553 18 0.375 10.8 2.48717 9.0699 19 0.35 75 12.6 2.57949 6.7312 20 0.4 75 14.42.61093 5.9152 21 0.375 35 13.5 2.81827 0.2868 22 0.425 35 15.3 2.82150.1957 23 0.375 45 13.5 2.81148 0.4779 24 0.425 45 15.3 2.80985 0.523725 0.375 55 13.5 2.78844 1.1230 26 0.425 55 15.3 2.77505 1.4955 27 0.37565 13.5 2.70996 3.2806 28 0.425 65 15.3 2.72913 2.7592 29 0.375 75 13.52.60633 6.0352 30 0.425 75 15.3 2.67199 4.3023

TABLE A. 20 Results of numerical simulation for the Carbon/epoxy pipeshaving 16 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 4 2.04884 19.0113 2 0.3 35 4.8 1.91739 21.6181 3 0.35 35 5.61.83941 23.0829 4 0.4 35 6.4 2.41512 10.8360 5 0.25 45 4 2.39079 11.42066 0.3 45 4.8 2.36811 11.9603 7 0.35 45 5.6 2.3956 11.3055 8 0.4 45 6.42.39885 11.2276 9 0.25 55 4 1.57573 27.5854 10 0.3 55 4.8 2.4052511.0739 11 0.35 55 5.6 2.37488 11.7997 12 0.4 55 6.4 2.47032 9.4876 130.25 65 4 2.38487 11.5620 14 0.3 65 4.8 2.39637 11.2871 15 0.35 65 5.62.38004 11.6770 16 0.4 65 6.4 2.40016 11.1962 17 0.25 75 4 2.3958311.3000 18 0.3 75 4.8 2.41153 10.9226 19 0.35 75 5.6 2.39003 11.4388 200.4 75 6.4 2.39868 11.2317

TABLE A. 21 Results of numerical simulation for the Carbon/epoxy pipeshaving 20 layers and angles between 50° and 60° Layer Winding TotalRebound Thickness Angle Thickness of Velocity Absorbed S. No. (mm)(degree) Plate (mm) (m/s) Energy (J) 1 0.25 50 5 2.37267 11.8522 2 0.350 6 2.39215 11.3881 3 0.35 50 7 2.41011 10.9568 4 0.4 50 8 2.484059.1475 5 0.25 52.5 5 2.44559 10.0954 6 0.3 52.5 6 2.40921 10.9785 7 0.3552.5 7 2.40164 11.1606 8 0.4 52.5 8 2.44105 10.2064 9 0.25 57.5 5 2.408510.9956 10 0.3 57.5 6 2.4444 10.1245 11 0.35 57.5 7 2.42232 10.6618 120.4 57.5 8 2.42097 10.6945 13 0.25 60 5 2.44265 10.1673 14 0.3 60 62.47024 9.4896 15 0.35 60 7 2.43268 10.4103 16 0.4 60 8 2.47767 9.3058

TABLE A. 22 Results of numerical simulation for the Carbon/epoxy pipeshaving 24 layers and angles between 50° and 60° Layer Winding TotalRebound Thickness Angle Thickness of Velocity Absorbed S. No. (mm)(degree) Plate (mm) (m/s) Energy (J) 1 0.25 50 6 2.38059 11.6640 2 0.350 7.2 2.46575 9.6004 3 0.35 50 8.4 2.49698 8.8255 4 0.4 50 9.6 2.481569.2093 5 0.25 52.5 6 2.39372 11.3505 6 0.3 52.5 7.2 2.48235 9.1897 70.35 52.5 8.4 2.4537 9.8968 8 0.4 52.5 9.6 2.50164 8.7090 9 0.25 57.5 62.40348 11.1164 10 0.3 57.5 7.2 2.41616 10.8109 11 0.35 57.5 8.4 2.444410.1245 12 0.4 57.5 9.6 2.45988 9.7450 13 0.25 60 6 2.43162 10.4361 140.3 60 7.2 2.43659 10.3151 15 0.35 60 8.4 2.4763 9.3397 16 0.4 60 9.62.5066 8.5848

TABLE A. 23 Results of numerical simulation for the Glass/epoxy pipeshaving 20 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 5 0 Penetrate 2 0.3 35 6 0 Penetrate 3 0.35 35 7 1.08274 34.13844 0.4 35 8 2.3477 12.4415 5 0.25 45 5 0 Penetrate 6 0.3 45 6 1.2259832.4849 7 0.35 45 7 2.33975 12.6278 8 0.4 45 8 2.37994 11.6794 9 0.25 555 2.34227 12.5689 10 0.3 55 6 2.32993 12.8571 11 0.35 55 7 2.3441812.5241 12 0.4 55 8 2.41696 10.7915 13 0.25 65 5 2.34617 12.4774 14 0.365 6 2.35908 12.1737 15 0.35 65 7 2.36987 11.9186 16 0.4 65 8 2.3781511.7220 17 0.25 75 5 0 Penetrate 18 0.3 75 6 2.31988 13.0908 19 0.35 757 2.36784 11.9667 20 0.4 75 8 2.40273 11.1344

TABLE A. 24 Results of numerical simulation for the Glass/epoxy pipeshaving 24 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 6 0 Penetrate 2 0.3 35 7.2 0.77370 37.0070 3 0.35 35 8.4 2.3465212.4692 4 0.4 35 9.6 2.35183 12.3445 5 0.25 45 6 1.12588 33.6620 6 0.345 7.2 2.33701 12.6919 7 0.35 45 8.4 2.37526 11.7907 8 0.4 45 9.62.37449 11.8090 9 0.25 55 6 2.33816 12.6650 10 0.3 55 7.2 2.3366512.7003 11 0.35 55 8.4 2.41873 10.7487 12 0.4 55 9.6 2.45415  9.8857 130.25 65 6 2.34628 12.4749 14 0.3 65 7.2 2.37862 11.7108 15 0.35 65 8.42.39118 11.4113 16 0.4 65 9.6 2.40827 11.0012 17 0.25 75 6 2.308713.3495 18 0.3 75 7.2 2.37866 11.7099 19 0.35 75 8.4 2.39493 11.3216 200.4 75 9.6 2.4412 10.2027

TABLE A. 25 Results of numerical simulation for the Glass/epoxy pipeshaving 28 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 7 1.18648 32.9613 2 0.3 35 8.4 2.34749 12.4465 3 0.35 35 9.82.34058 12.6084 4 0.4 35 11.2 2.35438 12.2845 5 0.25 45 7 2.3380612.6674 6 0.3 45 8.4 2.37643 11.7629 7 0.35 45 9.8 2.37471 11.8038 8 0.445 11.2 2.36973 11.9219 9 0.25 55 7 2.33516 12.7351 10 0.3 55 8.42.39867 11.2319 11 0.35 55 9.8 2.45515 9.8612 12 0.4 55 11.2 2.469859.4992 13 0.25 65 7 2.37526 11.7907 14 0.3 65 8.4 2.392 11.3917 15 0.3565 9.8 2.40366 11.1121 16 0.4 65 11.2 2.45935 9.7580 17 0.25 75 72.36673 11.9929 18 0.3 75 8.4 2.39012 11.4366 19 0.35 75 9.8 2.4453110.1023 20 0.4 75 11.2 2.42834 10.5158

TABLE A. 26 Results of numerical simulation for the Glass/epoxy pipeshaving 32 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 8 2.34847 12.4234 2 0.3 35 9.6 2.35175 12.3464 3 0.35 35 11.22.35126 12.3579 4 0.4 35 12.8 2.34264 12.5602 5 0.25 45 8 2.3824811.6189 6 0.3 45 9.6 2.37675 11.7553 7 0.35 45 11.2 2.37343 11.8342 80.4 45 12.8 2.45558 9.8506 9 0.25 55 8 2.37704 11.7484 10 0.3 55 9.62.4683 9.5375 11 0.35 55 11.2 2.46654 9.5809 12 0.4 55 12.8 2.511228.4689 13 0.25 65 8 2.37831 11.7182 14 0.3 65 9.6 2.41152 10.9229 150.35 65 11.2 2.46551 9.6063 16 0.4 65 12.8 2.5584 7.2729 17 0.25 75 82.40355 11.1147 18 0.3 75 9.6 2.4265 10.5605 19 0.35 75 11.2 2.4290610.4983 20 0.4 75 12.8 2.51793 8.3001

TABLE A. 27 Results of numerical simulation for the Glass/epoxy pipeshaving 36 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 9 2.34385 12.5318 2 0.3 35 10.8 2.34026 12.6159 3 0.35 35 12.62.31136 13.2881 4 0.4 35 14.4 2.7913 1.0432 5 0.25 45 9 2.3753 11.7897 60.3 45 10.8 2.38331 11.5992 7 0.35 45 12.6 2.44123 10.2020 8 0.4 45 14.42.79047 1.0664 9 0.25 55 9 2.43497 10.3546 10 0.3 55 10.8 2.46449 9.631411 0.35 55 12.6 2.50415 8.6462 12 0.4 55 14.4 2.7895 1.0934 13 0.25 65 92.4054 11.0703 14 0.3 65 10.8 2.45222 9.9331 15 0.35 65 12.6 2.557457.2972 16 0.4 65 14.4 2.61074 5.9202 17 0.25 75 9 2.39953 11.2113 18 0.375 10.8 2.42708 10.5464 19 0.35 75 12.6 2.5115 8.4618 20 0.4 75 14.42.53095 7.9715 21 0.375 35 13.5 2.46227 9.6861 22 0.425 35 15.3 2.798990.8283 23 0.375 45 13.5 2.79539 0.9290 24 0.425 45 15.3 2.79574 0.919225 0.375 55 13.5 2.51755 8.3097 26 0.425 55 15.3 2.79256 1.0080 27 0.37565 13.5 2.5793 6.7361 28 0.425 65 15.3 2.63406 5.3086 29 0.375 75 13.52.52931 8.0130 30 0.425 75 15.3 2.53922 7.7618

TABLE A. 28 Results of numerical simulation for the Glass/epoxy pipeshaving 40 layers Layer Winding Total Rebound Thickness Angle Thicknessof Velocity Absorbed S. No. (mm) (degree) Plate (mm) (m/s) Energy (J) 10.25 35 10 2.32956 12.8658 2 0.3 35 12 2.34196 12.5761 3 0.35 35 142.78597 1.1919 4 0.4 35 16 2.80272 0.7238 5 0.25 45 10 2.38761 11.4966 60.3 45 12 2.39204 11.3907 7 0.35 45 14 2.79567 0.9211 8 0.4 45 162.79667 0.8932 9 0.25 55 10 2.45024 9.9816 10 0.3 55 12 2.46826 9.538511 0.35 55 14 2.78827 1.1278 12 0.4 55 16 2.79438 0.9572 13 0.25 65 102.40469 11.0873 14 0.3 65 12 2.51472 8.3809 15 0.35 65 14 2.59576 6.310216 0.4 65 16 2.64831 4.9323 17 0.25 75 10 2.44626 10.0791 18 0.3 75 122.48627 9.0923 19 0.35 75 14 2.53047 7.9836 20 0.4 75 16 2.53942 7.7567

TABLE A. 29 Results of numerical simulation for the Glass/epoxy pipeshaving 24 layers with winding angles between 50° and 60° Layer WindingTotal Rebound Thickness Angle Thickness of Velocity Absorbed S. No. (mm)(degree) Plate (mm) (m/s) Energy (J) 1 0.25 50 6 2.31991 13.0901 2 0.350 7.2 2.33779 12.6737 3 0.35 50 8.4 2.3796 11.6875 4 0.4 50 9.6 2.4005611.1866 5 0.25 52.5 6 2.33659 12.7017 6 0.3 52.5 7.2 2.33785 12.6723 70.35 52.5 8.4 2.34672 12.4645 8 0.4 52.5 9.6 2.4401 10.2296 9 0.25 57.56 2.34376 12.5339 10 0.3 57.5 7.2 2.36829 11.9560 11 0.35 57.5 8.42.38649 11.5233 12 0.4 57.5 9.6 2.40847 10.9964 13 0.25 60 6 2.342412.5658 14 0.3 60 7.2 2.37386 11.8239 15 0.35 60 8.4 2.40307 11.1263 160.4 60 9.6 2.42099 10.6940

TABLE A. 30 Results of numerical simulation for the Glass/epoxy pipeshaving 28 layers with winding angles between 50° and 60 ° Layer WindingTotal Rebound Thickness Angle Thickness of Velocity Absorbed S. No. (mm)(degree) Plate (mm) (m/s) Energy (J) 1 0.25 50 7 2.34138 12.5897 2 0.350 8.4 2.37784 11.7294 3 0.35 50 9.8 2.40265 11.1364 4 0.4 50 11.22.38627 11.5286 5 0.25 52.5 7 2.33777 12.6742 6 0.3 52.5 8.4 2.3627712.0866 7 0.35 52.5 9.8 2.42132 10.6860 8 0.4 52.5 11.2 2.41214 10.90799 0.25 57.5 7 2.36438 12.0485 10 0.3 57.5 8.4 2.39016 11.4357 11 0.3557.5 9.8 2.42214 10.6662 12 0.4 57.5 11.2 2.44986 9.9909 13 0.25 60 72.37223 11.8626 14 0.3 60 8.4 2.38314 11.6032 15 0.35 60 9.8 2.4218310.6737 16 0.4 60 11.2 2.48559 9.1092

TABLE A. 31 Combinations for Top 2 layers of Woven Carbon/epoxy andResults for 55° filament wound pipes Carbon Glass Total Num- layer layerPlate ber thick- thick- Thick- of Rebound Absorbed ness ness ness Lay-Velocity Energy No. (mm) (mm) (mm) ers (m/s) (J) 1 0.25 0.25 20 52.34429 12.5215 2 0.3 0.3 20 6 2.3181 13.1320 3 0.35 0.35 20 7 2.3612812.1218 4 0.4 0.4 20 8 2.37573 11.7795

TABLE A. 32 Combinations for Top 4 layers of Woven Carbon/epoxy andResults for 55° filament wound pipes Carbon Glass Total Num- layer layerPlate ber thick- thick- Thick- of Rebound Absorbed ness ness ness Lay-Velocity Energy No. (mm) (mm) (mm) ers (m/s) (J) 1 0.25 0.25 20 52.33811 12.66621 2 0.3 0.3 20 6 2.33318 12.78136 3 0.35 0.35 20 7 2.372811.8491 4 0.4 0.4 20 8 2.37544 11.78642

TABLE A. 33 Combinations for Top 2 layers of Unidirectional Carbon/epoxyand Results for 55° filament wound pipes Carbon Glass Total Num- layerlayer Plate ber thick- thick- Thick- of Rebound Absorbed ness ness nessLay- Velocity Energy No. (mm) (mm) (mm) ers (m/s) (J) 1 0.25 0.25 20 52.34126 12.59251 2 0.3 0.3 20 6 2.34276 12.55738 3 0.35 0.35 20 72.36739 11.97732 4 0.4 0.4 20 8 2.3716 11.87757

TABLE A. 34 Combinations for Top 4 layers of Unidirectional Carbon/epoxyand Results for 55° filament wound pipes Carbon Glass Total Num- layerlayer Plate ber thick- thick- Thick- of Rebound Absorbed ness ness nessLay- Velocity Energy No. (mm) (mm) (mm) ers (m/s) (J) 1 0.25 0.25 20 52.34404 12.52738 2 0.3 0.3 20 6 2.33466 12.74681 3 0.35 0.35 20 72.48928 9.017425 4 0.4 0.4 20 8 2.50019 8.74525

NOMENCLATURE E11 Elastic Modulus in Longitudinal Direction [N/m²] E22Elastic Modulus in Transverse Direction [N/m²] E33 Elastic Modulus inTransverse Direction [N/m²] v12 Poisson's Ratio in plane containingfiber [Unitless] v13 Poisson's Ratio in plane containing fiber[Unitless] v23 Poisson's Ratio in transverse plane [Unitless] G12 ShearModulus in plane containing fiber [N/m²] G13 Shear Modulus in planecontaining fiber [N/m²] G23 Shear Modulus in transverse plane [N/m²] XtTensile strength in fiber direction [N/m²] Xc Compressive strength infiber direction [N/m²] Yt Tensile strength in transverse direction[N/m²] Yc Compressive strength in transverse direction [N/m²] S12In-Plane Shear Strength [N/m²] G_(f) ^(t) Fracture Toughness inlongitudinal tensile direction [J/m²] G_(f) ^(c) Fracture Toughness inlongitudinal compressive [J/m²] direction G_(m) ^(t) Fracture Toughnessin transverse tensile fracture [J/m²] mode G_(m) ^(c) Fracture Toughnessin transverse compressive [J/m²] fracture mode Gs In-Plane FractureToughness [J/m²] NSC Normalized Sensitivity Coefficient [Unitless] CFRPCarbon Fiber Reinforced Polymer GFRP Glass Fiber Reinforced Polymer

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What is claimed is:
 1. A method for predicting an impact resistance of acomposite material, the method comprising: designing an artificialneural network including a plurality of neurons; employing, by aprocessor, a sensitivity analysis to identify a parameter andquantitatively describe a degree of influence of the parameter on theimpact resistance of the composite material; training, by the processor,the artificial neural network to predict the impact resistance byadjusting an output of the plurality of neurons according to theparameter and the degree of influence identified in the employedsensitivity analysis; inputting data of the composite material into theartificial neural network; and utilizing the artificial neural networkto predict the impact resistance of the composite material.
 2. Themethod according to claim 1, wherein the artificial neural networkincludes: an input layer of neurons that receives data that is inputinto the artificial neural network; and an output layer of neurons thatoutputs a prediction of the impact resistance of the composite material.3. The method according to claim 1, wherein training the artificialneural network further includes: inputting sample data to the artificialneural network; measuring an error between known results of the sampledata and the prediction output from the artificial neural network; andreducing the error by applying a variable weighting factor to eachneuron of the plurality of neurons in the artificial neural network toadjust an output of each neuron.
 4. The method according to claim 3,wherein the error is a mean-squared error.
 5. The method according toclaim 1, wherein the input data of the composite material includes: astacking sequence of layers in the composite material; a layerthickness; a number of layers in the composite material; an orientationangle of the layers in the composite material; and a materialcomposition of the layers in the composite material.
 6. The methodaccording to claim 1, wherein the artificial neural network is a feedforward network.
 7. The method according to claim 2, wherein theartificial neural network further includes a hidden layer comprising aplurality of neurons, the hidden layer receives data output from theinput layer, and the hidden layer outputs processed data to the outputlayer.
 8. The method according to claim 7, wherein training theartificial neural network further includes: inputting sample data to theinput layer; measuring an error between known results of the sample dataand the prediction output from the output layer; and reducing the errorby: managing the hidden layer such that data output from neurons in theinput layer may be selected for input to individual neurons in thehidden layer, and applying a variable weighting factor to each neuron ofthe plurality of neurons in the artificial neural network to adjust anoutput of each neuron.
 9. A device for predicting an impact resistanceof a composite material, the device comprising: a processor configuredto: design an artificial neural network including a plurality ofneurons; employ a sensitivity analysis to identify a parameter andquantitatively describe a degree of influence of the parameter on theimpact resistance of the composite material; train the artificial neuralnetwork to predict the impact resistance by adjusting an output theplurality of neurons according to the parameter and the degree ofinfluence identified in the employed sensitivity analysis; input data ofthe composite material into the artificial neural network; and utilizethe artificial neural network to predict the impact resistance of thecomposite material.
 10. A system for predicting an impact resistance ofa composite material, the system comprising: the device according toclaim 9; and the artificial neural network.
 11. The device according toclaim 9, wherein the artificial neural network includes: an input layerof neurons that receives data that is input into the artificial neuralnetwork; and an output layer of neurons that outputs a prediction of theimpact resistance of the composite material.
 12. The device according toclaim 9, wherein training the artificial neural network furtherincludes: inputting sample data to the artificial neural network;measuring an error between known results of the sample data and theprediction output from the artificial neural network; and reducing theerror by applying a variable weighting factor to each neuron of theplurality of neurons in the artificial neural network to adjust anoutput of each neuron.
 13. A non-transitory computer readable mediumstoring computer readable instructions that when executed by a computercause the computer to perform a method comprising: designing anartificial neural network including a plurality of neurons; employing asensitivity analysis to identify a parameter and quantitatively describea degree of influence of the parameter on the impact resistance of thecomposite material; training the artificial neural network to predictthe impact resistance by adjusting an output of the plurality of neuronsaccording to the parameter and the degree of influence identified in theemployed sensitivity analysis; inputting data of the composite materialinto the artificial neural network; and utilizing the artificial neuralnetwork to predict the impact resistance of the composite material. 14.The non-transitory computer readable medium of claim 13, wherein theartificial neural network includes: an input layer of neurons thatreceives data that is input into the artificial neural network; and anoutput layer of neurons that outputs a prediction of the impactresistance of the composite material.
 15. The non-transitory computerreadable medium of claim 13, wherein training the artificial neuralnetwork further includes: inputting sample data to the artificial neuralnetwork; measuring an error between known results of the sample data andthe prediction output from the artificial neural network; and reducingthe error by applying a variable weighting factor to each neuron of theplurality of neurons in the artificial neural network to adjust anoutput of each neuron.
 16. The non-transitory computer readable mediumof claim 15, wherein the error is a mean-squared error.
 17. Thenon-transitory computer readable medium of claim 13, wherein the inputdata of the composite material includes: a stacking sequence of layersin the composite material; a layer thickness; a number of layers in thecomposite material; an orientation angle of the layers in the compositematerial; and a material composition of the layers in the compositematerial.
 18. The non-transitory computer readable medium of claim 13,wherein the artificial neural network is a feed forward network.
 19. Thenon-transitory computer readable medium of claim 14, wherein theartificial neural network further includes a hidden layer comprising aplurality of neurons, the hidden layer receives data output from theinput layer, and the hidden layer outputs processed data to the outputlayer.
 20. The non-transitory computer readable medium of claim 19,wherein training the artificial neural network further includes:inputting sample data to the input layer; measuring an error betweenknown results of the sample data and the prediction output from theoutput layer; and reducing the error by: managing the hidden layer suchthat data output from neurons in the input layer may be selected forinput to individual neurons in the hidden layer, and applying a variableweighting factor to each neuron of the plurality of neurons in theartificial neural network to adjust an output of each neuron.